Mathematical Models of Fluid Dynamics: Modelling, Theory, Basic Numerical Facts - An Introduction

Rainer Ansorge and Thomas Sonar Mathematical Models of Fluid Dynamics For additional information reagarding this topi...

Author: Rainer Ansorge | Thomas Sonar

16 downloads 406 Views 2MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!

Report copyright / DMCA form

DOWNLOAD PDF

Rainer Ansorge and Thomas Sonar Mathematical Models of Fluid Dynamics

For additional information reagarding this topic, please refer also to the following publications Durbin, P. A., Reif, B. A. P.

Statistical Theory and Modeling for Turbulent Flows 2001 ISBN 978-0-471-49744-8

Shivamoggi, B. K.

Theoretical Fluid Dynamics 1998 ISBN 978-0-471-05659-1

Middleman, S.

Introduction to Fluid Mechanics Principles of Analysis and Design 1998 ISBN 978-0-471-18209-2

Rainer Ansorge and Thomas Sonar

Mathematical Models of Fluid Dynamics Modeling, Theory, Basic Numerical Facts An Introduction Second, Updated Edition

WILEY-VCH Verlag GmbH & Co. KGaA

The Authors Prof. Dr. Rainer Ansorge Norderstedt, Germany [email protected] Prof. Dr. Thomas Sonar Institut Computational Mathematics Braunschweig, Germany [email protected]

All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate. Library of Congress Card No.: applied for British Library Cataloguing-in-Publication Data: A catalogue record for this book is available from the British Library. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliograﬁe; detailed bibliographic data are available on the Internet at http://dnb.d-nb.de. c 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form by photoprinting, microﬁlm, or any other means nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not speciﬁcally marked as such, are not to be considered unprotected by law. Typesetting le-tex publishing services oHG, Leipzig Printing Strauss GmbH, Mörlenbach Binding Litges & Dopf GmbH, Heppenheim Cover Spiesz Design, Neu-Ulm Printed in the Federal Republic of Germany Printed on acid-free paper ISBN 978-3-527-40774-3

Dedicated to our family members and to our students

VII

Contents

Preface to the Second Edition XI Preface to the First Edition XIII 1 1.1 1.2 1.3 1.4

Ideal Fluids 1 Modeling by Euler’s Equations 1 Characteristics and Singularities 10 Potential Flows and (Dynamic) Buoyancy 14 Motionless Fluids and Sound Propagation 29

2 2.1 2.2 2.3

Weak Solutions of Conservation Laws 33 Generalization of What Will Be Called a Solution 33 Trafﬁc Flow Example with Loss of Uniqueness 37 The Rankine–Hugoniot Condition 42

3 3.1 3.2 3.3 3.4

Entropy Conditions 49 Entropy in the Case of an Ideal Fluid 49 Generalization of the Entropy Condition 53 Uniqueness of Entropy Solutions 59 Kruzkov’s Ansatz 69

4 4.1 4.2 4.3

The Riemann Problem 73 Numerical Importance of the Riemann Problem The Riemann Problem for Linear Systems 75 The Aw–Rascle Trafﬁc Flow Model 77

5 5.1 5.2 5.3 5.4 5.5

Real Fluids 79 The Navier–Stokes Equations Model 79 Drag Force and the Hagen–Poiseuille Law 85 Stokes Approximation and Artiﬁcial Time 90 Foundations of the Boundary Layer Theory and Flow Separation Stability of Laminar Flows 102

73

95

VIII

Contents

104

5.6 5.7

Heated Real Gas Flows Tunnel Fires 106

6 6.1 6.2 6.3 6.4

Proving the Existence of Entropy Solutions by Discretization Procedures 113 Some Historical Remarks 113 Reduction to Properties of Operator Sequences Convergence Theorems 117 Example 120

7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9

Types of Discretization Principles 127 Some General Remarks 127 Finite Difference Calculus 131 The CFL Condition 135 Lax–Richtmyer Theory 136 The von Neumann Stability Criterion 141 The Modiﬁed Equation 144 Difference Schemes in Conservation Form 146 The Finite Volume Method on Unstructured Grids 148 Continuous Convergence of Relations 151

8 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9

A Closer Look at Discrete Models 155 The Viscosity Form 155 The Incremental Form 156 Relations 158 Godunov Is Just Good Enough 159 The Lax–Friedrichs Scheme 164 A Glimpse of Gas Dynamics 168 Elementary Waves 171 The Complete Solution to the Riemann Problem 178 The Godunov Scheme in Gas Dynamics 184

9 9.1 9.2 9.3 9.4 9.5 9.6

Discrete Models on Curvilinear Grids 187 Mappings 187 Transformation Relations 190 Metric Tensors 192 Transforming Conservation Laws 193 Good Practice 196 Remarks Concerning Adaptation 203

10 10.1 10.2 10.3 10.4

Finite Volume Models 205 Difference Methods on Unstructured Grids 205 Order of Accuracy and Basic Discretization 208 Higher-Order Finite Volume Schemes 209 Polynomial Recovery 211

114

Contents

10.5 10.6

Remarks Concerning Non-polynomial Recovery Remarks Concerning Grid Generation 218 Index 221 Suggested Reading 227

216

IX

XI

Preface to the Second Edition Because the ﬁrst edition of this introduction to a mathematical description of ﬂuid dynamics was so enthusiastically adopted by readers, the publishers encouraged the ﬁrst author to revise the ﬁrst edition and add any new results and considerations that are relevant to an introductory course. Because of this, the ﬁrst author introduced some new sections and remarks (e.g., Sections 4.3 , 5.6 , 5.7 , 7.9, etc.), and in turn encouraged the second author (who had already written parts of Chapter 7 for the ﬁrst edition) to widen his participation in the book and to contribute additional chapters touching upon some modern areas of the numerical treatment of ﬂuid ﬂows to this new edition. However, even in these new chapters, the introductory character of the book has been maintained. For more extensive representations of special areas of mathematical ﬂuid dynamics, like modeling, theory or numerical methods, we refer the reader to some monographs listed at the end of this book. Again, we thank the publishers for their support, our colleagues for fruitful discussions, and Mrs. Monika Jampert for technical help with respect to some LATEX problems. We also hope that readers of the new edition will again consider it useful for enriching their scientiﬁc interests and for executing their work.

Hamburg and Brunswick, September 2008

Rainer Ansorge Thomas Sonar

XIII

Preface to the First Edition Mathematical modeling is the process of representing problems from ﬁelds beyond mathematics itself using mathematics. The subsequent mathematical treatment of this model using theoretical and/or numerical procedures proceeds as follows: 1. Transition from the nonmathematical phenomenon to a mathematical description, which at the same time leads to the translation of problems formulated in terms of the original problem into mathematical problems. This task forces the scientist or engineer who intends to use mathematical tools to: – Cooperate with experts working in the ﬁeld that the original problem comes from. Thus, he/she has to learn the language of these experts and to understand their way of thinking (teamwork). – Create or accept an idealized description of the original phenomena, i.e., to ignore the properties of the original problem that are expected to be of no great relevance to the questions under consideration. These simpliﬁcations are useful since they reduce the complexity of both the model and its mathematical treatment. – Identify structures within the idealized problem and replace these structures with suitable mathematical structures. 2. Treatment of the mathematical substitute. This task normally requires: – Independent activity from the person working on the problem, who must work theoretically. – Treatment of the problem using tools from mathematical theory. – The solution of the particular mathematical problems that occur using these theoretical tools; in other words, differential equations or integral equations, optimal control problems, or systems of algebraic equations, etc., must be solved. Numerical procedures are often the only way to do this and to answer the particular questions of interest, at least approximately. The error in the approximate solution compared with the unknown

XIV

Preface to the First Edition

so-called exact solution does not normally affect the answer to the original problem a great deal, provided that the numerical method and tools applied are of sufﬁciently high accuracy. In this context, it should be realized that the quest for an exact solution does not make sense because of the idealizations mentioned above, and because the initial data presented with the original problem normally originate from statistics or from experimental measurements. 3. Retranslation of the results. The qualitative and quantitative statements obtained from the mathematical model then need to be retranslated into the language in which the original problem was formulated. In other words, the results must be interpreted with respect to their real-world meaning. This process again requires teamwork with the experts from the ﬁeld in which the problem originated. 4. Model checking. After retranslation, the results must be checked for relevance and accuracy, e.g., by performing experimental measurements. This work must be done by the experts from the ﬁeld in which the problem originated. If the mathematical results coincide sufﬁciently well with the results from experiments stimulated by the theoretical forecasts, the mathematical part is then completed, and a new tool that can be applied by the physicists, engineers, etc., to similar situations has been created. On the other hand, if the logical or computational errors are nontrivial, the model must be revised. In this situation, the gap between the results from the mathematical model and the real results can only have originated from using too much idealization during the modeling process. The development of mathematical models not only stimulates new experiments and leads to constructive prognostic (and hence technical) tools for physicists or engineers, but it is also important from the point of view of the theory of cognition: it allows us to understand the connections between different elements from an unstructured set of observations; in other words, to create theories. Mathematical descriptions have been used for centuries in various ﬁelds, such as physics, engineering, music, etc. Mathematical models are also used in modern biology, medicine, philology and economics, as well as in certain ﬁelds of art, like architecture or oriental ornaments. This book presents an introduction to models used in ﬂuid mechanics. Important properties of ﬂuid ﬂows can be derived theoretically from such models. We discuss some basic ideas for the construction of effective numerical procedures. Hence, all aspects of theoretical ﬂuid dynamics are addressed: modeling, mathematical theory and numerical methods. We do not expect the reader to be familiar with a lot of experimental work. A knowledge of some fundamental principles of physics, like conservation of mass,

Preface to the First Edition

conservation of energy, etc., is sufﬁcient. The most important idealization is (in contrast to the molecular structure of materials) the assumption of ﬂuid continua. The reader will ﬁnd the mathematics in the text easier to understand if he/she is acquainted with some basic elements of: – Linear algebra – Calculus – Partial differential equations – Numerical analysis – The theory of complex functions – Functional analysis Functional analysis only plays a role in the somewhat general theory of discretization algorithms described in Chapter 6. In this chapter, the question of the existence of weak entropy solutions of the problems under consideration is discussed. Physicists and engineers are normally not very interested in the treatment of this problem. Nevertheless, we felt that it should be included in this work so that this question not left unanswered. The lack of a solution immediately shows that a model does not ﬁt reality if there is a measurable course of physical events. Existence theorems are therefore important beyond the ﬁeld of mathematics alone. However, readers who are unacquainted with functional analytic terminology can of course skip this chapter. With respect to models and their theoretical treatment, as well as to the numerical procedures that occur in Sections 4.1, 5.3, 6.3 and Chapter 7, a brief introduction to mathematical ﬂuid mechanics as provided by this book can only present the most basic facts. However, the author hopes that this overview will generate interest in this ﬁeld among young scientists, and that it will familiarize people working in institutes and industry with some fundamental mathematical aspects. Finally, I wish to thank several colleagues for suggestions, particularly Thomas Sonar, who contributed to Chapter 7 when we organized a joint course for graduate students, 1) and Dr. Michael Breuss, who read the manuscript carefully. Last but not least, I thank the publishers, especially Dr. Alexander Grossmann, for their encouragement.

1) Parts of Chapters 1 and 5 are translations from parts of Sections 25.1, 25.2, 29.9 of: R. Ansorge, H. Oberle: Mathematik für Ingenieure, vol. 2, 2nd ed. Berlin: Wiley-VCH 2000.

XV

1

1 Ideal Fluids 1.1 Modeling by Euler’s Equations

Physical laws are mainly derived from conservation principles, such as conservation of mass, conservation of momentum, and conservation of energy. Let us consider a ﬂuid (gas or liquid) in motion, i.e., the ﬂow of a ﬂuid. 1) Let 0

1 u1 (x, y, z, t) B C u(x, y, z, t) = @u2 (x, y, z, t)A u3 (x, y, z, t)

be the velocity, 2) and denote by ρ = ρ(x, y, z, t) the density of this ﬂuid at point x = (x, y, z) and at time instant t. Let us take out of the ﬂuid at a particular instant t an arbitrary portion of volume W(t) with surface ∂W(t). The particles of the ﬂuid now move, and assume that W(t + h) is the volume formed at the instant t + h by the same particles that formed W(t) at time t. Moreover, let ϕ = ϕ(x, y, z, t) be one of the functions describing a particular state of the ﬂuid at time t at point x, such as mass per unit volume (= density), interior R energy per volume, momentum per volume, etc. Hence, W(t) ϕ d(x, y, z) gives the full amount of mass or interior energy, momentum, etc., of the volume W(t) under consideration. R We would like to ﬁnd the change in W(t) ϕ d(x, y, z) with respect to time, i.e., d dt

Z

(x, y, z, t) d(x, y, z) .

(1.1)

W(t)

1) Flows of other materials can be included too, e.g., the ﬂow of cars on highways, provided that the density of cars or particles is sufﬁciently high.

2) Note that bold letters in equations normally indicate vectors or matrices.

2

1 Ideal Fluids

We have d dt

Z W(t)

8 > Z 1< ϕ(x, y, z, t) d(x, y, z) = lim > h→0 h :

ϕ(˜y, t + h) d(y1, y2 , y3 )

W(t+h)

Z

– W(t)

9 > = ϕ(x, y, z, t) d(x, y, z) , > ;

where the change from W(t) to W(t + h) is obviously given by the mapping ˜y = x + h · u(x, t) + o(h) ` ´ ˜y = (y1 , y2 , y3 )T . The error term o(h) also depends on x but the property limh→0 h1 o(h) = 0 if differentiated with respect to space, provided that these spatial derivatives are bounded. The transformation of the integral taken over the volume W(t + h) to an integral over W(t) by substitution requires the integrand to be multiplied by the determinant of this mapping, i.e., by ˛ ˛ ˛(1 + h ∂x u1 ) h ∂ y u1 h ∂z u1 ˛˛ ˛ ˛ ˛ ` ´ ˛ ˛ 1 + h ∂ y u2 h ∂z u2 ˛ + o(h) ˛ h ∂ x u2 ˛ ˛ ˛ ˛ ˛ h∂ u (1 + h ∂ u )˛ h∂ u x 3

y 3

z 3

` ´ = 1 + h · ∂x u1 + ∂y u2 + ∂z u3 + o(h)

= 1 + h · div u(x, t) + o(h) . Taylor expansion of Vϕ(˜y, t + h) around (x, t) therefore leads to Z Z d ϕ(x, y, z, t) d(x, y, z) = {∂t ϕ + ϕ div u + u, ∇ϕ} d(x, y, z) . dt W(t)

(1.2)

W(t)

Here, ∇v denotes the gradient of a scalar function v, and · , · means the standard scalar product of two vectors out of R3 . The product rule from differentiation gives: ϕ div u + u, ∇ϕ = div(ϕ · u) , so that (1.2) leads to the so-called Reynolds’ transport theorem 3) Z Z d ϕ(x, y, z, t) d(x, y, z) = {∂t ϕ + div(ϕu)} d(x, y, z) . dt W(t)

(1.3)

W(t)

As already mentioned, the dynamics of ﬂuids can be described directly by conservation principles and – as far as gases are concerned – by an additional equation of state. 3) Osborne Reynolds (1842–1912); Manchester

1.1 Modeling by Euler’s Equations

1. Conservation of mass: If there are no sources or losses of ﬂuid within the subdomain of the ﬂow under consideration, the mass remains constant. Because W(t) and W(t+h) consist of the same particles, they have the same mass. R The mass of W(t) is given by W(t) ρ(x, y, z, t) d(x, y, z), and therefore d dt

Z

ρ(x, y, z, t) d(x, y, z) = 0 W(t)

must hold. Taking (1.3) into account (particularly for ϕ = ρ), this leads to the requirement Z {∂t ρ + div(ρu)} d(x, y, z) = 0 . W(t)

Since this has to hold for arbitrary W(t), the integrand has to vanish: ∂t ρ + div(ρu) = 0 .

(1.4)

This equation is called the continuity equation. 2. Conservation of momentum: Another conservation principle concerns the momentum of a mass, which is deﬁned as mass ~ velocity . Thus, Z

ρu d(x, y, z) W(t)

gives the momentum of the mass at time t of the volume W(t) and 0 1 q1 B C q = @q2 A = ρu q3

describes the density of momentum. The principle of the conservation of momentum, i.e., Newton’s second law force = mass ~ acceleration, then states that the change of momentum with respect to time equals the sum of all of the exterior forces acting on the mass of W(t). In order to describe these exterior forces, we take into account that there is a certain pressure p(x, t) at each point x in the ﬂuid at each instant t. If n is considered to

3

4

1 Ideal Fluids

be the unit vector normal on the surface ∂ W(t) of W(t), and it is directed outwards, the ﬂuid outside of W(t) acts on W(t) with a force given by Z – p n do (do = area element of ∂ W(t)) . ∂ W(t)

Besides the normal forces per unit surface area generated by the pressure, there are also tangential forces which act on the surface due to the friction generated by exterior particles along the surface. Though this so-called ﬂuid viscosity leads to a lot of remarkable phenomena, we are going to neglect this property at the ﬁrst step. Instead of real ﬂuids or viscous ﬂuids, we restrict ourselves in this chapter to so-called ideal ﬂuids or inviscid ﬂuids. This restriction to ideal ﬂuids, particularly to ideal gases, is one of the idealizations mentioned in the Preface. However, as well as exterior forces per unit surface area, there are also exterior forces per unit volume – e.g., the weight. Let us denote these forces per unit volume by k, such that Newton’s second law leads to Z Z Z d q d(x, y, z) = k(x, y, z, t) d(x, y, z) – p · n do . dt W(t)

∂W(t)

W(t)

Thus, by Gauss’ divergence theorem, we ﬁnd 0 Z 1 0Z 1 p n do ∂ p d(x, y, z) 1 x B C B C B∂W(t) C BW(t) C B C B C B Z C BZ C B C B C Z B C B p n2 doC = B ∂y p d(x, y, z) C C p n do = B B C B C B∂W(t) C BW(t) C ∂W(t) B Z C BZ C B C B C B C B C p n3 doA @ ∂z p d(x, y, z)A @ Z

∂W(t)

W(t)

∇p d(x, y, z) .

= W(t)

Together with (1.3), 8 9 0 1 > Z > div(q1 u) < = B C ∂t q + @div(q2 u)A – k + ∇p d(x, y, z) = 0 > > : ; div(q3 u) W(t) follows. Again, this has to be valid for any arbitrarily chosen volume W(t). If, moreover, div(qi u) = u, ∇qi + div u · qi is taken into account, ∂t q + u, ∇ q + div u · q + ∇p = k ,

1.1 Modeling by Euler’s Equations

i.e., ∂t q +

1 q, ∇ q + div ρ

„

1 q ρ

«

q + ∇p = k

(1.5)

has to be fulﬁlled. The number of equations represented by (1.5) equals the spatial dimension of the ﬂow, i.e., the number of components of q or u. By means of the continuity equation, (1.5) can be reformulated as ∂t u + u, ∇ u +

1 ∇p = kˆ , ρ

(1.6)

where the force k per unit volume has been replaced by the force ˆk = ρ1 k per unit mass. Equation (1.4) together with (1.5) or (1.6) are called Euler’s equations. 4) ρ, E, q1 , q2 , q3 are sometimes called conservative variables whereas ρ, ε, u1 , u2 , u3 are the primitive variables. Here, E := ρ ε +

ρ q2 u2 = ρ ε + 2 2ρ

gives the total energy per unit volume, where ε stands for the interior energy per unit mass, e.g., the heat per unit mass. ρ u2 2 obviously introduces the kinetic energy per unit volume. 5) u, ∇ u is called the convection term.

s

Remark Terms of the form ∂t w + w, ∇ w are often abbreviated in the literature to are called material time derivatives of the vector-valued function w.

Dw Dt

and

3. Conservation of energy: Next we consider the ﬁrst law of thermodynamics, namely: The change per time unit in the total energy of the mass of a moving ﬂuid volume equals the work done per time unit against the exterior forces. 4) Leonhard Euler (1707–1783); Basel, Berlin, St. Petersburg 5) · : 2-norm

5

6

1 Ideal Fluids

For ideal ﬂuids, this means that Z Z D Z E d ˆ u d(x, y, z) – E(x, y, z, t) d(x, y, z) = ρ k, p n, u do . 6) dt W(t)

∂W(t)

W(t)

The relation Z Z p n, u do = ∂W(t)

Z

p u, n do =

∂W(t)

div(p u) d(x, y, z) W(t)

follows from Gauss’ divergence theorem, so that (1.3) leads to Z n D Eo ˆ u ∂t E + div(E · u) + div(p · u) – ρ k, d(x, y, z) = 0 ,

∀ W(t) .

W(t)

If k can be neglected because of the small weight of the gas, or if kˆ is the weight of the ﬂuid per unit mass 7) and u is the velocity of ﬂow parallel to the Earth’s surface, we get „ « E+p ∂t E + div (1.7) q =0. ρ Explicitly written, and neglecting k, Eqs. (1.4), (1.5) and (1.7) become ∂t ρ + ∂x q 1 „

+ ∂y q 2

« 1 q 1 q 1 + p + ∂y ρ „ « 1 ∂t q 2 + ∂x + ∂y q2 q1 ρ „ « 1 + ∂y q3 q1 ∂t q 3 + ∂x ρ „ « E+p + ∂y q1 ∂t E + ∂x ρ

„

∂t q 1 + ∂x

+ ∂z q 3 1 q1 q2 ρ

«

„

+ ∂z

=0 1 q1 q3 ρ

«

=0

„

« „ « 1 1 =0 q 2 q 2 + p + ∂z q2 q3 ρ ρ „ « „ « 1 1 + ∂z q3 q2 q3 q3 + p = 0 ρ ρ „ « „ « E+p E+p + ∂z =0. q2 q3 ρ ρ

(1.8)

Hence, we are concerned with a system of equations that can be used to determine the functions ρ, q1 , q2 , q3 , E. However, we must note that there is an additional function that is sought, namely the pressure p. In the case of constant density 8) ρ, i.e., ∂t ρ = 0 ∀ (x, y, z), only four conservation variables have to be determined, so the ﬁve equations in (1.8) are sufﬁcient. Otherwise, particularly in the case of gas ﬂow, a sixth equation is needed, namely an 6) Work = force ~ length = pressure ~ area ~ length ⇒ work time = force ~ velocity = pressure ~ area ~ velocity 7) In other words, ˆk = g where g is the Earth’s gravitational acceleration. 8) The case of constant density is not necessarily identical to the case of an incompressible

ﬂow deﬁned by div u = 0, because in this case the continuity equation (1.4) is already fulﬁlled if the pair (ρu) shows the property ∂t ρ + u∇ρ = 0, which does not necessarily imply ρ = const.

1.1 Modeling by Euler’s Equations

equation of state. State variables of a gas are: T = temperature , p = pressure , ρ = density , V = volume , ε = energy/mass , S = entropy/mass , and a theorem of thermodynamics says that each of these state variables can be uniquely expressed in terms of two of the other state variables. Such relations between three state variables are called equations of state. Thus, p can be expressed by ρ and ε (and hence by ρ, E, q); for inviscid so-called γ-gases, the relation is given by: „ « q2 p = (γ – 1) ρε = (γ – 1) E – (1.9) 2ρ with γ = const > 1, such that only the functions ρ, q, E have to be determined. c Here, the adiabatic exponent γ is the ratio cVp of the speciﬁc heats. In the case of air, we have γ W 1.4. Using vector-valued functions, systems of differential equations can also be described by a single differential equation. In this way, and taking (1.9) into account as an additional equation, (1.8) can be written as ∂t V + ∂x f1 (V ) + ∂y f2 (V ) + ∂z f3 (V ) = 0

(1.10)

with V = (ρ, q1 , q2 , q3 , E)T and with

0

q1

1

B C B1 C B q q + pC Bρ 1 1 C B C B C B 1 C B C q q f1 (V ) = B ρ 2 1 C , B C B 1 C B q3 q1 C B C B ρ C B C @ E+p A q1 ρ

0

q2

1

B C B 1 C B C q q B ρ 2 1 C B C B C B1 C B q2 q2 + p C f2 (V ) = B ρ C, B C B 1 C B q3 q2 C B C B ρ C B C @ E+p A q2 ρ

0

q3

1

B C B 1 C B C q q B ρ 3 1 C B C B C B 1 C B C q q f3 (V ) = B ρ 3 2 C . B C B1 C B q q + pC B C 3 3 Bρ C B C @ E+p A q3 ρ

The functions fj (V ) are called ﬂuxes. If J f1 , J f2 , J f3 are the Jacobians, (1.10) becomes ∂t V + J f1 (V ) · ∂x V + J f2 (V ) · ∂y V + J f3 (V ) · ∂z V = 0 .

(1.11)

Because of their particular meaning in physics, systems of differential equations of type (1.10) are called systems of conservation laws, even if they do not arise from physical aspects. Obviously, (1.11) is a quasilinear system of ﬁrst-order partial differential equations. Such systems must be deﬁned by initial conditions V(x, 0) = V0 (x) , where V0 (x) is a prescribed initial state, and by boundary conditions.

(1.12)

7

8

1 Ideal Fluids

If the ﬂow does not depend on time, it is called a stationary or steady-state ﬂow, and initial conditions do not occur. Otherwise, the ﬂow is termed nonstationary. As far as ideal ﬂuids are concerned, it can be assumed that the ﬂuid ﬂow is tangential along the surface of a solid body 9) ﬁxed in space or along the bank of a river, etc. In this case, u, n = 0

(1.13)

is one of the boundary conditions, where n are the outward-directed normal unit vectors along the surface of the body. There are often symmetries with respect to space such that the number of unknowns can be reduced, e.g., in the case of rotational symmetry combined with polar coordinates. If it is found that there is only one spatial coordinate, the problem is termed one-dimensional. 10) If rectangular space variables are used and x is the only one that remains, we end up with the system ∂t V + J f (V ) · ∂x V = 0 with

0 0 1 ρ B C V = @ qA , E

(1.14)

q

1

B1 C B 2 C B q + pC f (V ) = B ρ C B C @ E+p A q ρ

and with the equation of state „ « q2 p = (γ – 1) E – . 2ρ Hence, 0

0

B B B 3 – γ q2 B – J f(V ) = B B 2 ρ2 B B @ Eq q3 (γ – 1) 3 – γ 2 ρ ρ

1 (3 – γ) γ

0 q ρ

E 3(γ – 1) q2 – ρ 2 ρ2

1

C C C C γ – 1C . C C C q A γ ρ

As can easily be veriﬁed, λ1 = ρq is a solution of the characteristic equation of J f (V), namely of j ﬀ q q2 E – λ3 + 3 λ2 – (γ2 – γ + 6) 2 – γ(γ – 1) λ ρ 2ρ ρ „ 2 « 3 γ q Eq γ – γ(γ – 1) 2 = 0 . – +1 + 2 2 ρ3 ρ 9) For example, a wing 10) Two- or three-dimensional problems are deﬁned analogously.

1.1 Modeling by Euler’s Equations

The other two eigenvalues of J f (V ) are then the roots of 2q q2 E λ + (γ2 – γ + 2) 2 – γ(γ – 1) = 0: ρ 2ρ ρ s q2 q E q2 = ± + γ(γ – 1) – (γ2 – γ + 2) 2 2 ρ ρ ρ 2ρ

λ2 –

λ2,3

q = ± ρ q = ± ρ

s

γ(γ – 1) s

E q2 – γ(γ – 1) 2 ρ 2ρ

(1.15)

» – r γ(γ – 1) q2 q p E– = ± γ . ρ 2ρ ρ ρ

Thus, these eigenvalues are real and different from each other (p > 0). Deﬁnition If all the eigenvalues of a matrix A(x, t, V ) are real and if the matrix can be diagonalized, the system of equations ∂t V + A(x, t, V ) ∂x V = 0 is termed hyperbolic at (x, t, V). If the eigenvalues are real and different from each other such that A can deﬁnitely be diagonalized, the system is said to be strictly hyperbolic. Obviously then, (1.14) is strictly hyperbolic for all (x, t, V) under consideration. q

By the way, because ρq = u, the velocity of the ﬂow, and because the local sound velocity ˆc (cf. (1.66)), the eigenvalues are: λ1 = u ,

λ2 = u + ˆc ,

γp ρ

describes

λ3 = u – ˆc ,

and have equal signs in the case of supersonic ﬂow, whereas the subsonic ﬂow is characterized by the fact that one eigenvalue has a different sign to the others. Deﬁnition Let u(x, t0 ) be the velocity ﬁeld of the ﬂow at instant t0 and let x0 be an arbitrary point from the particular subset of R3 which is occupied by the ﬂuid at this particular instant. If, at this moment, the system ˆ

˜ x (s), u(x(s), t0 ) = 0 11)

of ordinary differential equations with the initial condition x(0) = x0 has a unique solution x = x(s) for each of these points x0 , each of the curves x = x(s) is called a streamline at instant t0 .

9

10

1 Ideal Fluids

Here, the streamlines may be parametrized by the arc length s with s = 0 at point x0 , such that x , x = 1 . Thus, the set of streamlines at an instant t0 shows a snapshot of the ﬂow at this particular instant. It does not necessarily describe the trajectories along which the ﬂuid particles move over time. Only if the ﬂow is a stationary one, i.e., if u, ρ, p, ˆk are independent of time, do the streamlines and trajectories coincide: a particle moves along a ﬁxed streamline over time.

1.2 Characteristics and Singularities

As an introductory example for a more general investigation of conservation laws, let us consider the scalar Burgers’ equation 12) without exterior forces: „ « 1 2 ∂t v + ∂x (1.16) v = 0 , (x, t) ∈ Ω , i.e., x ∈ R , t v 0 , 2 which is often studied as a model problem from a theoretical point of view and also as a test problem for numerical procedures. Here, the ﬂux is given by f (v) = 12 v 2 . If v0 (x) = 1 –

x 2

(1.16a)

is chosen as a particular example of an initial condition, the unique and smooth solution 13) turns out to be 2–x , v(x, t) = 2–t but this solution only exists locally, namely for 0 u t < 2; as time increases it runs into a singularity at t = 2. As a matter of fact, classical existence and uniqueness theorems for quasilinear ﬁrst-order partial differential equations with smooth coefﬁcients only ensure the unique existence of a classical smooth solution in a certain neighborhood of the initial manifold, provided that the initial condition is also sufﬁciently smooth. The occurrence of discontinuities or singularities does not depend on the smoothness of the ﬂuxes: assume that v(x, t) is a smooth solution of the problem ∂t v + ∂x f(v) = 0

for x ∈ R ,

tv0

v(x, 0) = v0 (x) 11) Here, [·, ·] is the vector product in R3 . 12) J. Burgers: Nederl. Akad. van Wetenschappen 43 (1940) 2–12.

13) In other words, v is continuously differentiable.

1.2 Characteristics and Singularities

t

v

v v

v

Fig. 1.1 Formation of discontinuities.

in a certain neighborhood immediately above the x-axis. Obviously, this solution is constant along the straight line x(t) = x0 + t f (v0 (x0 ))

(1.17)

that crosses the x-axis at x0 , where x0 ∈ R is chosen arbitrarily. Its value along this line is therefore v(x(t), t) = v0 (x0 ). This can easily be veriﬁed by d v(x(t), t) = ∂t v(x(t), t) + ∂x v(x(t), t) · x (t) dt = ∂t v(x, t) + ∂x v(x, t) · f (v0 (x0 )) = ∂t v(x, t) + ∂x v(x, t) · f (v(x, t)) = ∂t v + ∂x f(v) = 0 . The straight lines of (1.17), each of which belong to a particular x0 , are called the characteristics of the given conservation law. In the case of x0 < x1 , but for example 0 < f (v0 (x1 )) < f (v0 (x0 )), 14) the characteristic through (x0 , 0) intersects the characteristic through (x1 , 0) at an instant t1 > 0, so that at the point of intersection the solution v must have the value v0 (x0 ) as well as the value v0 (x1 ) =/ v0 (x0 ). Therefore, a discontinuity will occur at the instant t1 or even earlier. With respect to ﬂuid dynamics, one notable type of discontinuity is shocks (discussed later). When applied to Burgers’ equation (1.16) with an initial function of (1.16a), the characteristic through a point x0 on the x-axis is given by x – x0 t=2 , 2 – x0 such that the discontinuity we found at t = 2 can also be immediately understood via Fig. 1.2. If systems of conservation laws are considered instead of the scalar case, i.e., V(x, t) ∈ Rm , m ∈ N , ∀ (x, t) ∈ Ω , and if only one spatial variable x occurs, a system of characteristics is deﬁned as follows: 14) Hence, v0 (x0 ) = / v0 (x1 ) too

11

12

1 Ideal Fluids

Deﬁnition If V(x, t) is a solution of (1.10), in the case of only one space variable x, the oneparameter set x(i) = x(i) (t, χ)

(χ ∈ R : set parameter)

of real curves deﬁned by the ordinary differential equation x˙ = λi V(x, t)

(1.18)

for every ﬁxed i (i = 1, . . . , m) is called the set of i-characteristics of the particular system that belongs to V. Here, λi V ) (i = 1, · · · , m) are the eigenvalues of the Jacobian J f (V ). Obviously, this deﬁnition coincides in the case of m = 1 with the previously presented deﬁnition of characteristics. Let us ﬁnally – using an example – study the situation for a system of conservation laws if the system is linear with constant coefﬁcients: ∂ tV + A ∂ xV = 0

(1.19)

with a constant Jacobian (m, m)-matrix A. Moreover, let us assume the system to be strictly hyperbolic. From (1.18), the characteristics turn out to be the set of straight lines given by x(i) (t) = λi t + x(i) (0) ,

(i = 1, . . . , m) ,

and are independent of V. Hence, for every ﬁxed i, the characteristics belonging to this set are parallel. If S = (s1 , s2 , . . . , sm ) is the matrix whose columns consist of the eigenvectors of the Jacobian, and if Λ denotes the diagonal matrix consisting of the eigenvalues of A, A = SΛS–1

(1.20)

follows.

v v

v Fig. 1.2 Discontinuity of the solution to Burgers’ equation.

1.2 Characteristics and Singularities

ˆ are introduced by If new variables V ˆ = S–1 V , V the system takes the form ˆ + Λ ∂xV ˆ =0, ∂tV

ˆ 0 = S–1 V0 . V

(1.21)

This is a decoupled system: ˆ ˜ ∂t vˆi + λi ∂x vˆi = 0 , vˆi (x, 0) = S–1 V0 (x) i = vˆi0 (x) ,

(i = 1, . . . , m) .

Each of the equations of this system is an independent scalar equation called an advection equation. The solution is vˆi (x, t) = vˆi0 (x – λi t)

(i = 1, . . . , m) .

(1.22)

Hence, the state at instant t moves with velocity λi in the positive or negative x-direction according to the sign of λi . This is called wave propagation, where the velocity of propagation is described by λi . Obviously, 0 1 vˆ1 B . C C V = (s1 , . . . , sm ) B @ .. A = vˆ1 s1 + vˆ2 s2 + . . . + vˆm sm . vˆm Each of the vector-valued functions V(i) (x, t) := vˆi (x, t) · si solves the system of differential equations because of ∂t V(i) + SΛS–1 ∂x V(i) = ∂t vˆi si + ∂x vˆi λi si = (∂t vˆi + λi ∂x vˆi ) si = 0 . V(i) (i = 1, . . . , m) is often called the solution belonging to the i-th set of characteristics and to the given initial value vˆi0 (x). Obviously, the vector functions V(i) (i = 1, . . . , m) are linearly independent.

s

Remark The fact that a sufﬁciently smooth solution of a nonlinear initial value problem often only exists in the neighborhood of the initial manifold, while the corresponding real-world process exists globally, means that this solution is not accepted by the physicist or engineer. The mathematician is asked to ﬁnd a global solution. This forces the mathematician to create a more general deﬁnition of the solution such that the real-world situation can be described in a satisfactory way. Suitable deﬁnitions of weak solutions will be presented in Chapter 2.

13

14

s

1 Ideal Fluids

Remark As far as the scalar linear problem ∂t v + ∂x (a v) = 0 ,

a = const ,

(1.23)

is concerned, v often describes a concentration, and the ﬂux is then simply given by f(v) = a v. Many physical processes include a further ﬂux of the particular form –ε ∂x v

(ε > 0) ,

proportional to the drop in concentration. The transport phenomenon is then mathematically modeled by ∂t v + ∂x (a v – ε ∂x v) = 0 ,

i.e.,

∂t v + a ∂x v = ε ∂xx v .

(1.24)

Because of the diffusion term on the right hand side, this equation is of parabolic type. Parabolic equations yield smoother and smoother solutions as time t increases. Therefore, shocks will be smeared out as soon as diffusion occurs. This effect of parabolic equations can easily be demonstrated using examples like the following one. The sum v(x, t) = e 2 (x– 2 t) a

a

∞ X ν=1

(–1)ν+1

sin(νx) –ν2 t e ν 2

converges uniformly for t > 0 because of the factors e–ν t . This also holds after multiple termwise differentiations of this sum with respect to t as well as with respect to x so that v is a sufﬁciently smooth function for all x ∈ R , t > 0. It solves (1.24) 15) according to the classical understanding, but leads for t = 0 to the function a

v0 (x) = e 2 x · w(x) where w(x) represents the Fourier expansion of the 2π-periodic discontinuous function (x for –π < x < π 2 w(x) = . 0 for x = ±π The curve described by w(x) is sometimes called a saw blade curve.

1.3 Potential Flows and (Dynamic) Buoyancy

Let us now try to investigate the forces acting on solid bodies 16) dipped into a ﬂuid ﬂow at a ﬁxed position. For convenience, we restrict ourselves to stationary ﬂows of 15) In the case of ε = 1 16) For example, on the wings of an aircraft

1.3 Potential Flows and (Dynamic) Buoyancy

inviscid ﬂuids. Moreover, it will be assumed that the magnitudes of the velocities are such that the density can be regarded as a constant. Thus, the ﬂow is incompressible and the partial derivatives with respect to t occurring in (1.8) vanish. The ﬁrst equation in (1.8) – if written in primitive variables, i.e., the continuity equation (1.4) – then reduces to div u = 0 ,

(1.25)

` ´T where 17) u = u1 (x, y, z) , u2 (x, y, z) , u3 (x, y, z) again denotes the velocity vector of the ﬂow at the space position x = (x, y, z)T . The second, third, and fourth equations of (1.8) formulated by means of primitive variables could be written as the vector-valued Euler equation (1.6), and they lead in the case of a stationary ﬂow to

u, ∇ u +

1 ∇p = kˆ . ρ

(1.26)

Let us also assume that the ﬂow is irrotational; this means that the circulation I (1.27) Z := u(x) dx C

vanishes for every closed contour C within every simply connected subdomain of the ﬂow area. Then u can be derived from a potential φ; i.e., there is a scalar function φ such that u = ∇φ ,

(1.28)

curl u = 0

(1.29)

and

holds within this area. By the way, the vector curl u is often called the vorticity vector or angular velocity vector, and a trajectory of a ﬁeld of vorticity vectors is called a vortex line. Because of (1.28), a ﬂow of this type is called a potential ﬂow, and φ is the so-called velocity potential. The ﬁfth equation in (1.8) can be omitted assuming that knowledge of the energy density E is of no interest. From (1.29), the relation u, ∇ u = 12 ∇(u2 ) – [u, curl u] leads together with (1.26) to „ « 1 1 (1.30) u2 + p = ˆk . ∇ 2 ρ 17) For the time being, z denotes the third space variable; it will later denote the complex variable x + i y, but we have taken care to avoid any confusion.

15

16

1 Ideal Fluids

Formula (1.30) shows that a ﬂuid ﬂow of the particular type under consideration, namely an approximately stationary, inviscid, incompressible, and irrotational ﬂow can only exist if the exterior forces are the gradient of a scalar function. In other words, these forces must be conservative, i.e., there must be a potential Q with kˆ = –∇Q such that (1.30) leads to „

∇

1 1 u2 + p + Q 2 ρ

«

=0.

i.e., to 1 1 u2 + p + Q = const . 2 ρ

(1.31)

Equation (1.31) is called the Bernoulli equation, 18) and is none other than the energy conservation law for this particular type of ﬂow. p is called the static pressure, whereas the term ρ2 u2 , i.e., the kinetic energy per volume, is often called the dynamic pressure. As far as incompressible ﬂow in a circular pipe is concerned, the velocity will necessarily increase as soon as the diameter of the pipe decreases, and – because of (1.31) – this will lead to decreasing pressure within the narrow part of the pipe. This phenomenon is called the hydrodynamic paradox. Applications include carburetors and jet streams.

s

Remark A necessary condition for the existence of irrotational ﬂow was the conservative character of the exterior forces. Let us now assume that these forces are conservative instead, i.e., ˆk = –∇Q. Moreover, let us allow the ﬂow to be compressible, with a particular dependence ρ = ρ(p), termed barotropic ﬂow. Integration of (1.26) along a streamline from a constant point P0 to a variable point P then leads to ZP j P0

1 ∇(u2 ) + [curl u, u] + ∇Θ + ∇Q 2

with Z

Θ :=

dp , ρ(p)

hence ∇Θ =

1 ∇p , ρ

18) Daniel Bernoulli (1700–1782); St. Petersburg, Basel

ﬀ

ds = 0 ,

1.3 Potential Flows and (Dynamic) Buoyancy

and with ds =

1 u ds u

(s = arc length) .

Because P was arbitrary, and because of [curl u, u], u = 0, this result leads to the generalized Bernoulli equation 1 u2 + Θ + Q = const , 2 which in the case of constant density, i.e., Θ = ρp , seems to coincide completely with (1.31). However, it should be noted that the constant on the right hand side can now change from streamline to streamline. Additionally, we ﬁnd ﬁ ﬂ I I d d dx d u(x) dx = u, ds Z= dt dt dt ds C C I I du dx + u du = dt C C I I du dx = {u, ∇ u} dx = dt C C ﬀ I j 1 ∇p + ∇Q dx =– ρ C I ` ´ = – ∇ Θ + Q dx = 0 . C

This is Kelvin’s theorem, 19) which says that the circulation along a closed curve in an inviscid barotropic ﬂow does not change over time. If an irrotational ﬂow with an arbitrary type of incompressibility is considered, (1.28) leads together with (1.25) to Δφ = 0

(1.32)

whereas, for a compressible ﬂuid in the case of stationary ﬂow, the continuity equation (1.4) is div(ρu) = 0 . Let us additionally assume the ﬂow to be barotropic. Then, because of div(ρu) = ρ div u +

dρ ∇p, u dp

19) Lord Kelvin of Largs (1824–1907); Glasgow

17

18

1 Ideal Fluids

and because of s dp ˆc = dρ

(cf. (1.67)) ,

(1.26) leads to „ “ ”2 « „ “ ”2 « „ “ ”2 « u1 u2 u3 1– ∂xx φ + 1 – ∂yy φ + 1 – ∂zz φ ˆc ˆc ˆc “u u ” u2 u 3 u3 u 1 1 2 · ∂ y u1 + 2 · ∂ z u2 + 2 · ∂ x u3 = 0 , –2 2 ˆc ˆc ˆc

(1.33)

as far as the exterior forces vanish. In this situation, (1.33) generalizes (1.32). Obviously, (1.33) is a quasilinear partial differential equation of second order for φ, which is certainly elliptic if M :=

u <1, ˆc

(1.34)

i.e., in areas of subsonic ﬂow. Deﬁnition M is called the Mach number. 20) In particular, if we consider a constant ﬂow in the x-direction which is only disturbed in the neighborhood of a slim airfoil 21) with a small angle of attack, the products of the values ui (i = 2, 3) with each other and with u1 can be neglected when compared with 1. This leads to u2 = u21 and to a shortened version of (1.33), namely to „ “ ” « u1 2 1– ∂xx φ + ∂yy φ + ∂zz φ = 0 . (1.35) ˆc This equation is hyperbolic in areas of supersonic ﬂow (M > 1), and the full equation (1.33) is also hyperbolic in this case. Let us extend our idealizations by assuming that the ﬂow under consideration is a two-dimensional plane ﬂow. This means that one of the components of the velocity vector u in a rectangular coordinate system, e.g., the component in the z-direction, vanishes for all (x, y) ∈ R2 and for all t v 0: u3 = 0 .

(1.36)

In the case of a two-dimensional supersonic ﬂow along a slim airfoil for which (1.35) holds, Mach’s angle β, determined from |sin β| =

1 <1, M

20) Ernst Mach (1838–1916); Graz, Praha, Vienna 21) A cross-section of a wing or another rigid body in a plane parallel to the direction of the ﬂow

1.3 Potential Flows and (Dynamic) Buoyancy

Fig. 1.3 Flow around a wing if the angle of attack is small.

Fig. 1.4 Mach’s net in the case of a linearized supersonic ﬂow (M = const > 1).

describes the angle between the characteristics of the wave equation (1 – M2 )∂xx φ + ∂yy φ = 0 and the ﬂow direction given by the direction of the x-axis. The set of all of these characteristics is called Mach’s net. This net plays an important role when so-called methods of characteristics are used in order to establish efﬁcient numerical procedures. The two-dimensional case of rotational symmetry, e.g., ﬂow along a projectile, leads correspondingly to Mach’s cone. In the two-dimensional plane situation, (1.29) becomes 0 1 –∂z u2 B C curl u = @ ∂z u1 A = 0 . ∂ x u2 – ∂ y u1 Thus, u1 and u2 are independent of the third spatial variable z: 0 1 u1 (x, y) B C u = @u2 (x, y)A 0

19

20

1 Ideal Fluids

with ∂ x u2 – ∂ y u1 = 0 .

(1.37)

After introducing the vector 0 1 –u2 (x, y) B C v := @ u1 (x, y)A 0

(1.38)

and using the continuity equation (1.25), we obtain 0

1 0 1 0 0 B C B C curl v = @ 0 A=@ 0 A=0, ∂ x u1 + ∂ y u2 div u

such that v can also be derived from the potential in simply connected parts of the ﬂuid area. In other words, there is a scalar function ψ = ψ(x, y) with –u2 = ∂x ψ ,

u1 = ∂ y ψ .

(1.39)

ψ is called the stream function. (1.28) leads to ∂x φ = ∂y ψ ,

∂y φ = –∂x ψ .

(1.40)

Obviously, (1.40) can be interpreted as the system of Cauchy–Riemann equations of the complex function Ω(z) := φ(x, y) + i ψ(x, y)

(1.41)

depending on the complex variable z = x + i y. Ω is called the complex velocity potential of the plane potential ﬂow under consideration.

Fig. 1.5 Linearized supersonic ﬂow around a slim rotatory cone.

1.3 Potential Flows and (Dynamic) Buoyancy

We are going to assume that the ﬁrst partial derivatives of the functions φ and ψ are continuous such that Ω is found to be a holomorphic function, and we intend to study the forces acting on rigid bodies when dipped into such a ﬂuid ﬂow. Because we reduced reality to a plane ﬂow, the rigid body is assumed to be very long with respect to the direction of the third spatial variable; more precisely, it must be of inﬁnite length from the point of view of mathematics. Therefore, the ﬂow around the contour of the cross-section of the body within the (x, y)-plane, e.g., around the contour of an airfoil of a long wing, is of interest. H We stated that the circulation Z = C u dx vanishes as long as the closed contour C is the boundary of a simply connected domain within the ﬂuid area in the case of a potential ﬂow. Because of our plane model, C represents a simple closed contour in the (x, y)-plane, i.e., in R2 . Now, consider the situation where the boundary Γ of a rigid body is dipped into the ﬂuid, e.g., a wing. If its airfoil is part of the area surrounded by C, this interior domain of C is no longer a simply connected domain of the ﬂuid area. Hence, the circulation Z around the airfoil does not necessarily vanish but is found to fulﬁll I I u dx = u dx . C

Γ

In order to proof this relation, we take into account that Γ and C can, in a ﬁrst step, be connected by two auxiliary lines in such a way that two simply connected domains G1 and G2 , with contours C1 and C2 , respectively, will occur. Because of curl u = 0 in G1 as well as in G2 , we see in a second step that I I u dx = 0 as well as u dx = 0 C1

C2

holds. This leads to I I u dx + u dx = 0 C1

C2

(cf. Fig. 1.6). We realize in a third step that the integrations back and forth along each of the auxiliary lines extinguish each other such that I I I I u dx + u dx = u dx + u dx C1

C2

C

–Γ

Fig. 1.6 Auxiliary step in the computation of the buoyancy generated by plane potential ﬂows.

21

22

1 Ideal Fluids

results. Here, –Γ denotes the contour of the rigid body along the negative direction, and this ends the proof.

s

Remark It should be noted that the proof given here is simpliﬁed because Γ not only passes through the interior of the domain occupied by the ﬂuid but is also part of its boundary. As a matter of fact, one must ﬁrst, roughly speaking, investigate the case where Γ is replaced by a line Γε of distance ε from Γ that passes only through the ﬂuid, and in a next step one must study the limit situation ε → 0. Let Γ now be parametrized by r = r(τ) , 0 u τ u T. Then ZT

ZT

˙ dτ (1.42) u(r(τ)), r˙ (τ) dτ = {u1 x˙ + u2 y} ! 0 x(τ) ˙ holds, and r˙ = is a vector tangential to the curve Γ at the point (x(τ), y(τ)) y˙ (τ) such that ! 1 y˙ n := p (1.43) x˙ 2 + y˙ 2 –x˙ Z=

0

is a normal unit vector of Γ at this point. According to (1.13), we assume that the velocity of the ﬂow at the surface of the rigid body is tangential to this surface: u, n = 0 along Γ , ˙ From this, and with (1.42), the circulation becomes i.e., u1 y˙ = u2 x. ZT

(u1 – i u2 ) (x˙ + i y˙ ) dτ ,

Z= 0

i.e., Z

Z=

w(z) dz ,

(1.44)

Γ

with w(z) := u1 (x, y) – i u2 (x, y) .

(1.45)

w(z) is a holomorphic function in the whole domain outside the airfoil because (1.39) yields ∂x u1 = ∂xy ψ = ∂yx ψ = ∂y (–u2 ) , and because (1.28) leads to ∂y u1 = ∂xy φ = ∂yx φ = –∂x (–u2 ) ,

1.3 Potential Flows and (Dynamic) Buoyancy

the Cauchy–Riemann equations for the function w(z) are fulﬁlled. Hence, also with respect to the computation of the circulation, integration along C instead of Γ is permitted: Z (1.46) Z = w(z) dz . C

Here, we choose C in such a way that it lies in the annulus between two concentric circles, where circle Kr of radius r surrounds the airfoil and KR is a circle with a sufﬁciently great arbitrary radius R > r. Without any loss of generality, we assume the center of each circle to be the origin of the former (x, y)-plane, which now becomes the complex z-plane. w(z) can be represented in the annulus (and therefore for every z ∈ C in particular) by a Laurent series 22) around the center z0 = 0 (cf. Fig. 1.7): w(z) =

+∞ X

aν zν .

ν=–∞

Fig. 1.7 Annulus around an airfoil.

We know from experience that the ﬂuid ﬂow is often only inﬂuenced by the rigid body within the neighborhood (which has a certain size) of the contour; the ﬂow around a ship that crosses a calm lake at a constant velocity provides an example of this. From this point of view, we are going to assume that the velocity u = u(x, y) is constant for |z| → ∞: ! u1,∞ lim u(x, y) = (1.47) u2,∞ |z|→∞ with constant components u1,∞ and u2,∞ such that lim w(z) = u1,∞ – i u2,∞ .

|z|→∞

22) Pierre Alphonse Laurent (1813–1854); Le Havre

23

24

1 Ideal Fluids

Because the coefﬁcients aν (ν = 0, ±1, ±2, . . .) are constant numbers which do not depend on R, and because R is allowed to tend to inﬁnity, we ﬁnd for |z| → ∞ that all the aν with positive indices must vanish. Hence, the result lim w(z) = a0 = u1,∞ – i u2,∞

(1.48)

|z|→∞

follows. On the other hand, Cauchy’s formula a–1 =

1 2π i

I

w(ζ) dζ C

leads to Z = 2π i a–1 .

(1.49)

The (two-dimensional) force K to be calculated, which is caused by the ﬂow and acts on the rigid body, is found to be:

K=

K1 K2

!

ZL

p n ds

=–

[K] = ˆ

0

[force] ˜ length

!

,

(1.50)

where n is the unit normal vector from (1.43), p denotes the pressure along the surface of the body caused by the ﬂuid, s measures the arc length along Γ beginning at an arbitrary point on it, and where L is the total length of Γ. Let us study this force separately from the other exterior forces acting on the body; i.e., let us assume that the sum of these other forces vanishes. This leads to a constant potential Q in Bernoulli’s equation (1.31) and therefore to a constant total pressure p0 :=

ρ u2 + p . 2

Because Γ is a closed contour, Therefore,

K=–

0

n ds = 0. 23)

ZL n ZL o ρ ρ p0 – u2 n ds = u2 n ds , 2 2 0

23)

RL

0

ds p 2 2 = x˙ + y˙ , hence dτ ZL

ZT p

n ds = 0

0

1 x˙ 2

+

y˙ 2

„ « „ « „ « y(T) – y(0) 0 y˙ ds dτ = = –x(T) + x(0) 0 –x˙ dτ

1.3 Potential Flows and (Dynamic) Buoyancy

i.e., ρ K1 = 2 K2 = –

ZT p 0

ρ 2

1 x˙ 2

ZT p 0

+ y˙ 2

u2 y˙

ds dτ , dτ

1 ds u2 x˙ dτ . dτ x˙ 2 + y˙ 2

The temporary introduction of the complex number k = K2 + i K1

(1.51)

yields k=–

ρ 2

ZT 0

1 ds ρ p ˙ u2 (x˙ – i y) dτ = – 2 2 dτ 2 x˙ + y˙

ZT

u2 (x˙ – i y˙ ) dτ . 0

The complex number within the parentheses on the right hand side of ` ´ ˙ = (u1 – i u2 ) (u1 + i u2 ) (x˙ – i y) ˙ u2 (x˙ – i y˙ ) = u21 + u22 (x˙ – i y)

= w(z) (u1 x˙ + u2 y˙ + i u2 x˙ – i u1 y˙ ) is not really complex because the imaginary part vanishes (cf. (1.13)). It can therefore be replaced by its conjugate complex number: u2 (x˙ – i y˙ ) = w(z) (u1 x˙ + u2 y˙ – i u2 x˙ + i u1 y˙ ) ˙ = w2 (z) z˙ . = w(z) (u1 – i u2 ) (x˙ + i y) This leads to ρ k=– 2

ZT

w2 (z) z˙ dτ = – 0

ρ 2

Z

w2 (z) dz . Γ

Because w is holomorphic, w2 is also a holomorphic function. Hence, Z ρ w2 (z) dz . k=– 2 C

However, within the annulus we have “ ”“ ” a–1 a–2 a–1 a–2 a0 + + 2 + ... + 2 + ... w2 (z) = a0 + z z z z a0 a–1 A–2 A–3 2 = a0 + 2 + 2 + 3 + ... z z z with certain coefﬁcients Aν (ν = –2, –3, . . .).

(1.52)

25

26

1 Ideal Fluids

Cauchy’s residuum formula and (1.52) therefore lead to ρ k = – 2a0 a–1 · 2π i . 2 Thus, (1.48) and (1.49) yield k = –ρ (u1,∞ – i u2,∞ ) Z , and because the circulation Z is real (cf. (1.27)), comparison of the real and imaginary parts of (1.51) results in K1 K2

= ρ u2,∞ Z = –ρ u1,∞ Z .

(1.53)

In particular, if u2,∞ = 0 but u1,∞ =/ 0, i.e., if the undisturbed ﬂow is parallel to the x-axis, and if Z =/ 0, we obtain a lift K2 =/ 0, i.e., there is a force acting on the rigid body perpendicular to the direction of the ﬂow. By using an appropriate wing construction, the airfoils can yield Z < 0, so that an aircraft can lift its own weight, a hydrofoil can rise out of the water, etc. Of course, Archimedes’ static buoyancy, as given by (1.4), must also be taken into account.

s

Remark The formulae (1.53) are called Kutta–Zhukovsky buoyancy formulae. 24)

s

Remark Whereas the ﬁrst equation in (1.53) describes the buoyancy at least qualitatively in a correct manner, the result K1 = 0 contradicts reality. Also, in the case of an incompressible irrotational stationary ﬂow (as more or less realized by calmly ﬂowing streams), the ﬂow will apply some force to the rigid body (e.g., a bridge pier) parallel to the direction of ﬂow. Of course, this contradiction results from one of our idealizations: the assumption of an inviscid and therefore frictionless ﬂuid. In order to understand what really happens parallel to the ﬂow, we must reduce the amount of idealization. This will later be achieved by using the so-called Navier– Stokes equations rather than the Euler equations, and the so-called no-slip condition given below rather than (1.13). u = 0 along Γ .

(1.54)

Equation (1.54) expresses the idea that moving viscous ﬂuids leave a monomolecular stationary layer on impermeable walls of solid bodies because of adhesion. Thus, friction along such surfaces does not mean friction between the ﬂuid and the solid material, but rather friction between ﬂuid particles. Of course, there are also other boundary conditions where partial slip occurs on the surface, e.g., in rareﬁed gas ﬂow, with porous walls, etc. The tangential component of the ﬂow is then proportional to the local shear stress. 24) Martin Wilhelm Kutta (1867–1927); Stuttgart; Nikolai Jegorowitsch Zhukovsky (1847–1921); Moscow

1.3 Potential Flows and (Dynamic) Buoyancy

If (1.33) is reduced to the case of an irrotational, stationary, plane ﬂow (i.e., u3 = 0, ∂y u1 = ∂x u2 ), if there are no exterior forces, and if (u1 , u2 ) is for convenience replaced with (u, v), we get: „ “ ”2 « „ “ ” « ´ u v 2 uv ` 1– ∂x u + 1 – ∂y v – 2 · ∂y u + ∂x v = 0 . (1.55) ˆc ˆc ˆc The uv-plane is called the hodograph plane. We assume the equations u = u(x, y) v = v(x, y) to be invertible such that x and y can be expressed by u and v, i.e., ˛ ˛ ˛ ∂ u ∂ u˛ ˛ x y ˛ D := ˛ ˛ =/ 0 . ˛ ∂ x v ∂y v ˛ In other words, it will be assumed that the vectors ∂x u and ∂y u are linearly independent. This leads immediately to ∂x u = D ∂v y ,

∂y u = –D ∂v x ,

∂x v = –D ∂u y ,

∂y v = D ∂u x .

The nonlinear equation (1.55) together with the equation ∂y u = ∂x v for the functions u(x, y) and v(x, y) of the irrotational ﬂow can therefore – after division by D – be transformed into the linear equations „ « „ « v2 u2 uv 1 – 2 ∂u x + 1 – 2 ∂v y + 2 (∂v x + ∂u y) = 0 ˆc ˆc ˆc ∂v x – ∂u y = 0

(1.56)

for the functions x(u, v) and y(u, v). The transition from the original equations to the linear equations (1.56) is called the method of hodographs and corresponds to the Legendre transformation in the theory of partial differential equations. In simply connected domains, x(u) can be derived from a potential Θ (i.e., x = Θu , y = Θv ), because of curl x(u) = (0, 0, ∂u y – ∂v x)T = 0 . Equation (1.56) therefore yields „ « „ « v2 u2 uv 1 – 2 ∂uu Θ + 1 – 2 ∂vv Θ + 2 2 ∂uv Θ = 0 . ˆc ˆc ˆc If polar coordinates (w = u , α) are used in the hodograph plane, i.e., u = w cos α ,

v = w sin α ,

1 ∂u = cos α ∂w – sin α ∂α , w 1 ∂v = sin α ∂w + cos α ∂α , w

(1.57)

27

28

1 Ideal Fluids

Equation (1.57) becomes the so-called hodograph equation: „ « „ « 1 w2 1 w2 1 – 2 ∂w Θ = 0 ∂ww Θ + 2 1 – 2 ∂αα Θ + ˆc ˆc w w

(1.58)

which does not explicitly contain α. If we try to solve (1.58) by the ansatz Θ(w, α) = g(w) sin(mα)

or Θ(w, α) = g(w) cos(mα) ,

(m ∈ R) ,

(1.59)

we obtain for the unknown function g(w) the ordinary differential equation „ « „ « 1 w2 m2 w2 1 – 2 g (w) – 2 1 – 2 g(w) = 0 . (1.60) g

(w) + ˆc ˆc w w Solutions of the hodograph equation of type (1.59) are called Chapligin solutions. 25) Let gm (w) be a solution of (1.60) that belongs to a particular m and let Θm (w, α) be the solution of (1.59) that corresponds to this solution. Examples: m = 0 leads to g

0 1 w =– + 2 , g 0 w ˆc hence g 0 =

(0)

c1 12 w22 e ˆc . w 1 w2

The power series of e 2 ˆc2 converges uniformly for all values of w. Integration can therefore be performed term-by-term, yielding ( ) ∞ ` w ´2ν X (0) (0) ˆc g0 (w) = c1 ln w + + c2 . ν+1 νν! 2 ν=1 (0)

(0)

Here, c1 and c2 are arbitrary constants. Analogously for m = 1: ( ) ∞ 1 w ln w X w2(ν+1) (1) (1) g1 (w) = c1 w + c2 + + . w2 2ˆc2 2ν+2ˆc2(ν+2) ν=0 Because (1.58) is linear and homogeneous, all linear combinations of the particular Chapligin solutions also solve Eq. (1.58). The coefﬁcients of the expansion as (m) (m) well as the constants c1 and c2 must be chosen in such a way that the expansion ﬁts the given situation, at least approximately. 25) C.A. Chapligin: Sci. Ann. Univ. Moscow. Math. Phys. 21 (1904) 1–121

1.4 Motionless Fluids and Sound Propagation

1.4 Motionless Fluids and Sound Propagation

Obviously, because viscosity only plays a role in moving ﬂuids, the results of this section are also valid for real ﬂuids. In a ﬁrst step, let us consider the case for constant density ρ, which is approximately realized in liquids. In this situation, (1.26) leads for motionless ﬂuids (i.e., for u = 0) to the so-called hydrostatic equation „ « p kˆ = ∇ , ρ and (1.31) becomes Q+

p = const . ρ

Let us assume that we do not yet know how the free surface of a motionless liquid behaves if only the force of gravity and a constant exterior (e.g., atmospheric) pressure p0 affects this liquid. Hence, the force per mass unit is given by kˆ = (0, 0, –g)T

(g = acceleration due to gravity) ,

such that Q = g z + const (i.e., ρ g z + p = const). In particular, if (x, y, z0 ) is a point on the surface, ρ g z + p = ρ g z0 + p0 or ρ g (z – z0 ) = –(p – p0 ) = –ˆp

(1.61)

holds. Here, pˆ is the overpressure inside the liquid compared with the exterior pressure. Because of z0 =

const – p0 , ρg

z0 is constant, and thus independent of (x, y). In other words, the surface of the liquid is a plane, or more precisely it is parallel to the Earth’s surface. If h = z0 – z is the height of an arbitrarily shaped liquid column, (1.61) gives ρ g h q = pˆ q , where q is the base area of the column. The force affecting this base is given by the right hand side of the equation and does not depend on the form of the column, whereas the left hand side gives the weight of a cylindrical column of the ﬂuid with the same base area and the same height. This phenomenon is called the hydrostatic paradox. We are now going to dip a solid body of volume V and surface F into a stationary liquid.

29

30

1 Ideal Fluids

Fig. 1.8 Hydrostatic paradox.

Obviously, an overall force K affects the body, where Z K = (K1 , K2 , K3 )T = – p n do + G . F

Here, p is the interior pressure of the liquid, G is the weight of the body, and n denotes the outward-directed normal unit vector at the points on the surface. Because of G = (0 , 0 , –G)T , (1.61) yields K1 = ρ g

« Z „ Z p0 z – z0 – n1 do = ρ g a, n do ρg F

F

p0 , 0, 0)T . The divergence theorem therefore leads to ρg Z Z ∂(z – z0 ) div a dV = ρ g K1 = ρ g dV = 0 . ∂x

with a := (z – z0 –

V

V

Analogously, K2 = 0. K3 is found to be given by Z

K3 = ρ g V

∂(z – z0 ) dV – G = ρ g V – G , ∂z

so that K = (0, 0, ρ g V – G)T .

(1.62)

ρ is the density of the liquid (!), so ρ g V is the weight of the particular part of the liquid which is displaced by the solid body. Hence, the body is affected by a force directed against the direction of action of its weight, and this (static) buoyancy equals the weight of the displaced quantity of the liquid (Archimedes’ principle). 26) In order to study the propagation of sound in a ﬂuid, we assume that the ﬂuid does not move from a macroscopic point of view. Moreover, we will only consider the sound effects resulting from very small changes in the density and pressure 26) Archimedes (about 220 BC); Syracuse

1.4 Motionless Fluids and Sound Propagation

within the ﬂuid. Therefore, we no longer assume the density to be constant, but we assume only small variations in it. Let ρˆ and pˆ be the averages of the density and the pressure, respectively. Only these averages are expected to be constant quantities. The density disturbances will be expressed by ` ´ ρ=ρ ˆ 1 + σ(x, t)

where |σ| << 1 and the spatial derivatives of σ are small. 27) The continuity equation (1.4) then becomes ` ´ ˆ div (1 + σ)u = 0 , ρ ˆ σt + ρ

and so, after dividing by ρ ˆ and taking the assumptions for σ into account, σt + div u = 0 .

(1.63)

Experiments show that the propagation of sound waves occurs more or less adiabatically, i.e., without any gain or loss of heat. The equation of state to be taken into account is therefore ρ–γ p = const 28)

(1.64)

such that p = pˆ

„ «γ ρ = (1 + σ)γ . ρ ˆ

Because σ is small, (1 + σ)γ W 1 + γ σ . Hence p = pˆ(1 + γ σ) , so that 1 pˆ γ ∇p = ∇σ . ρ (1 + σ) ρ ˆ

(1.65)

In our model of small disturbances, the velocity of the ﬂuid particles and its spatial derivatives are so small that higher-order terms of these quantities can be neglected 27) Our results do not necessarily hold in the case of the propagation of large variations in the pressure or density, such as can occur in the case of detonations. 28) γ from (1.9)

31

32

1 Ideal Fluids

compared with ﬁrst-order terms. In other words, the convection term in (1.6) can be neglected. Also, exterior forces do not play a role in our scenario. If (1.65) is then put into (1.6), we ﬁnd ut = –

γ pˆ ∇σ , ρ ˆ

where 1 + σ was approximated by 1. Forming the divergence of this term, and then changing the sequence of the time derivative and the spatial derivatives, we end up with (div u)t = –

γ pˆ Δσ . ρˆ

This result can be compared with (1.63) when differentiated with respect to t. This comparisons yields σtt =

γ pˆ Δσ , ρ ˆ

a wave equation for σ. It shows that the sound waves propagate within the ﬂuid with the velocity s

ˆc =

γ pˆ . ρˆ

(1.66)

Deﬁnition ˆc is called the local speed of sound.

s

Remark Because of (1.64), ˆc can also be represented by s

ˆc =

dp . dρ

(1.67)

33

2 Weak Solutions of Conservation Laws 2.1 Generalization of What Will Be Called a Solution

As already announced, in this chapter we are going to discuss the introduction of a suitable globalization of the deﬁnition of a conservation law solution. We already know that such a solution cannot be expected to be a smooth function, because we noted that discontinuities can even arise from smooth initial states. Moreover, discontinuities that arise at an instant t0 can move with respect to space and time, as will be seen later. Of course, this also holds if the initial state already shows one or more discontinuities. On the other hand, in a suitable deﬁnition of a global solution, discontinuities that existed at an instant t0 can also disappear for t > t0 . Examples of this will be presented later. All of these phenomena can occur in scalar problems as well as in systems of conservation laws. We restrict ourselves within this chapter to only one space variable, but allow the sought solution V = (v1 , . . . , vm )T to consist of more than one component. Hence, we are going to treat the problem ∂t V + ∂x f (V ) = 0 on Ω = {(x, t)|x ∈ R, t v 0} , V(x, 0) = V0 (x) ,

(2.1)

where we will assume for the time being that f is at least one time differentiable. Example Another example of this kind – besides (1.14) – is the system of the so-called shallow-water equations. These equations describe the shape and the velocity of surface waves for shallow liquids. Here, we consider an idealized situation, where waves of an inviscid incompressible liquid travel along a shallow canal in a direction that coincides with the x-axis, and the ﬂow of liquid particles in the z-direction can be neglected. This idealization appears acceptable if the amplitudes of the surface waves are small compared with the wavelength. The width b of the canal is assumed to be constant, and we are looking for the height h(x, t) of the liquid surface above the bottom of the canal (cf. Fig. 2.1).

34

2 Weak Solutions of Conservation Laws

Fig. 2.1 Shallow water ﬂow along a canal.

The mass of the ﬂuid within a segment x u ξ u x + Δx of the canal at instant t is given by x+Δx Z

m(t) = bρ

h(ξ, t) dξ . x

With (1.3), if ϕ is speciﬁcally chosen as ϕ = bρh, conservation of mass leads to d d m(t) = bρ dt dt =0,

x+Δx Z

x+Δx Z

{∂t h + ∂x (uh)} dξ

h(ξ, t) dξ = bρ x

x

∀ (x, Δx) ,

i.e., ∂t h + ∂x (uh) = 0 . This replaces the continuity equation in this particular situation. The momentum of the mass m(t) reads x+Δx Z

bρ

u(ξ, t)h(ξ, t) dξ . x

The force per length unit arising from the hydrostatic pressure, namely Zh

(h – z)ρg dz = ρg

bp = 0

h2 2

(g: acceleration due to gravity) ,

acts on the liquid in the segment considered here. The principle of the

2.1 Generalization of What Will Be Called a Solution

conservation of momentum then leads via (1.3), now with ϕ = bρuh, to ∂t (ρhu) + ∂x (ρhu2 ) + ∂x p = 0 , i.e., to „ « h2 ∂t (hu) + ∂x hu2 + g = 0 . 2

This equation turns out to be Euler’s equation for our example. If the continuity equation and the Euler equation are multiplied by g; if ∂t (hu) is replaced by ∂t hu + h∂t u; if a variable ψ(x, t) := gh(x, t) is introduced; if the continuity equation is again taken into account; and ﬁnally, if the second equation is divided by ψ, we obtain the system of shallow-water equations: ! ! uψ ψ ∂t + ∂x u2 =0. u +ψ 2 Obviously, this is in fact a particular system of the type corresponding to (1.2). In order to now establish a suitable generalized deﬁnition of a weak solution, let us temporarily assume that there is a smooth solution to problem (2.1) on Ω. Put this solution into (2.1), then multiply (2.1) by an arbitrary test function Φ ∈ C10 (Ω). Here, C10 (Ω) is deﬁned to be the set of all functions that are continuously differentiable on Ω and have compact support. In particular, each of the functions Φ vanishes at the boundary of its support, with the possible exception of the parts of the boundary that belong to the x-axis. Now we integrate by parts over Ω. In fact, this is only an integration over the compact support of Φ, and hence over a closed, bounded region. This integration leads to 29) Z Ω

Z+∞ {V∂t Φ + f (V )∂x Φ} d(x, t) + V0 (x)Φ(x, 0) dx = 0,

∀ Φ ∈ C10 (Ω) .

–∞

(2.2) Obviously, the left hand side of (2.2) no longer makes sense for only sufﬁciently smooth functions V; it also holds for all V ∈ Lloc 1 (Ω), i.e., for all vector-valued functions V whose components can be integrated over compact subsets of Ω in the sense of Lebesgue. Let us now forget the way we obtained (2.2), and instead ask for functions V ∈ Lloc 1 (Ω) which fulﬁll (2.2). These functions will be called weak solutions of the original problem (2.1), and these solutions are no longer necessarily smooth. 29) It should be noted that this reformulation of the problem uses the conservation form of the differential equation and cannot be generally applied to arbitrary quasilinear partial differential equations.

35

36

2 Weak Solutions of Conservation Laws

On the other hand, our motivation for (2.2) shows that every smooth solution is also a weak solution. Conversely, every weak solution V which is a smooth function in a neighborhood D(P0 ) of a point P0 = (x0 , t0 ) not only fulﬁlls (2.2) but also (2.1) according to the classical understanding. Indeed, let Φ ∈ C10 (Ω) be an arbitrary test function with support D(P0 ). Then Z∞

Z

{V∂t Φ + f (V )∂x Φ} d(x, t) + D(P0 )

V0 (x)Φ(x, 0) dx = 0

–∞

holds and leads to Z {∂t V · Φ – ∂t (VΦ) + ∂x f (V ) · Φ – ∂x ( f (V ) · Φ)} d(x, t) – D(P0 )

Z∞

V0 (x)Φ(x, 0) dx = 0 . –∞

By the divergence theorem, the last equation can be written as Z

*

Z

{∂t vi + ∂x fi (V )} Φ d(x, t)

– ∂D(P0 ) Z∞

D(P0 )

! +! ` ´ fi (V ) ,n Φ d ∂D(P0 ) vi

v0,i (x)Φ(x, 0) dx = 0

– –∞

(i = 1, . . . , m) , where n denotes the outward normal unit vector on the boundary ∂D(P0 ) of D(P0 ) and where ·, · means the standard R2 -scalar product. If ∂D(P0 ) does not contain points on the x-axis both boundary integrals vanish, because we have Φ = 0 along ∂D(P0 ). Otherwise, the ﬁrst boundary integral turns into an integral over the particular part of the boundary that belongs to the x-axis because Φ vanishes along the rest of this boundary. In each of these cases we ﬁnd Z {∂t vi + ∂x fi (V )} Φ d(x, t) D(P0 )

Z∞ (*

– –∞

! !+ ) fi (V ) 0 , + v0,i Φ(x, 0) dx = 0 , vi –1

i.e., for i = 1, . . . , m , Z

{∂t V + ∂x f (V )} Φ(x, t) d(x, t) + D(P0 )

Z∞ {V(x, 0) – V0 (x)}Φ(x, 0) dx = 0 .

–∞

2.2 Trafﬁc Flow Example with Loss of Uniqueness

Since Φ was arbitrary and V was assumed to be smooth, this can only hold if ∂t V + ∂x f (V ) = 0 on D(P0 ), and hence particularly at P0 , and if V(x0 , 0) = V0 (x0 ). However, because P0 was chosen arbitrarily in its neighborhood, our assertion follows. This concept of a weak solution of (2.1) yields the advantage that the set of admissible solutions can be extended considerably. In particular, discontinuous solutions can be admitted so long as the Lebesgue integrability is not disturbed. However, there is also a signiﬁcant disadvantage: the uniqueness of the solution, which was at least guaranteed locally (along with its existence) is lost. There is often more than one weak solution, so a new question arises: how can we select the particular weak solution that answers the original real-world problem from among the set of all of the the weak solutions of (2.1)? In fact, it is often only possible to describe the solution or weak solution of a differential equation problem approximately, via a numerical procedure. Numerical procedures normally consist of discrete models of the original problem, i.e., in ﬁnite-dimensional problems whose solutions are expected to lie in a certain neighborhood of the unknown solution of the original problem. In particular, we expect the approximate solution to converge to this unknown solution if the dimension of the discrete model problem tends to inﬁnity. If convergence of the numerical solution to a weak solution of the original problem can be shown, one must ensure that this weak solution coincides with the relevant solution instead of one of the other weak solutions. 30) Let us now study an example where the transition to a weak formulation really does lead to a loss of solution uniqueness.

2.2 Trafﬁc Flow Example with Loss of Uniqueness

As far as sufﬁciently dense car trafﬁc is concerned, the density ρ (cars/mile) as well as the ﬂux f (cars/h) in macroscopic trafﬁc ﬂow models are approximately regarded as functions deﬁned on a continuum, e.g., for a u x u b. The spatial variable x (miles) denotes the positions along a one-way lane which is interpreted as a onedimensional set, and 0 u t < ∞. If there are no approaches or exits and no crossings along the part of the road under consideration, the conservation of mass (i.e., the constant number of cars) obviously leads within our continuous model to the demand for the validity of the continuity equation ∂t ρ + ∂x f = 0 30) Of course, convergence is a basic requirement and must be accompanied by other important properties of the numerical procedure in order to make it an applicable method.

(2.3)

37

38

2 Weak Solutions of Conservation Laws

with f = vρ ,

(2.4)

where v = v(x, t) (miles/h) denotes the speed of the cars at position x and at instant t. Equation (2.3) must also take an initial condition ρ(x, 0) = ρ0 (x)

(2.5)

with a given function ρ0 as well as a spatial boundary condition, since the part of the road under consideration cannot be regarded as a lane of inﬁnite length. Moreover, it should be assumed that – hopefully! – there is no friction between cars. From experience, ρ explicitly depends mainly on the speed of the cars: ρ = ρ(v) .

(2.6)

This is an empirically given decreasing and strictly monotonic function. Hence, we can also consider v to be a function of ρ. Let vF be the mean value of the individual maximal car velocities if the road in front of a driver is more or less empty. This value is assumed to be a constant, e.g., vF = 80 miles/h. Obviously, the initial density ρ0 (x) = ρ(v(x, 0)) is known as soon as the initial speed distribution v(x, 0) = v0 (x)

(2.7)

is given. Instead of ρ(v) or of v(ρ), the graph of f = v(ρ) · ρ =: f (ρ)

(2.8)

is often given empirically and turns out to be a strictly concave function called the fundamental diagram. Thus, this trafﬁc ﬂow model leads to an initial value problem for a scalar conservation law of type (2.1), where V consists of just the one component of ρ such that f (V ) = f(ρ). Let ρ∗ denote the jam concentration, i.e., the density of cars at a standstill in a congestion. This value is also assumed to be a constant, similar to vF , e.g., ρ∗ = 400 cars/ml. A very simple f(ρ) model, later used in order to explain certain facts and connections by example, was given by Greenshields: „ « ρ f(ρ) = vF · ρ · 1 – ∗ . (2.9) ρ

2.2 Trafﬁc Flow Example with Loss of Uniqueness

Thus, the ﬂux vanishes – in other words, there is (almost) no trafﬁc – if the trafﬁc density vanishes or if the road users are at a standstill within a trafﬁc jam.´ ` ∗ ∗ Equation (2.9) is a concave parabola that has its maximum at ρ2 , vF4ρ , and it leads – together with (2.4) – to the relation „ « v ρ = ρ∗ 1 – . vF

Hence, in the case of Greenshields’ model, the graph of ρ(v) is found to be a straight line. The more general model for arbitrary ﬂuxes is due to Lighthill & Whitham. 31) Using Greenshields’ model, we are going to begin at the start of the trafﬁc jam, i.e., (

ρ0 (x) =

ρ∗

for

x<0

0

for

xv0

(2.10)

for t0 = 0. The start of the trafﬁc jam (red trafﬁc light) is located at x = 0, and we identify the forward direction of the trafﬁc with the positive direction along the x-axis.

s

Remark Conservation law problems with initial values that are constant on the left hand side of a discontinuity as well as on its right hand side are called Riemann problems. 32) With (2.9) and (1.17), the characteristics are given by 1 (x – x0 ) = vF

(

–t

for

x0 < 0

t

for

x0 v 0

.

Obviously, the characteristics do not intersect for t > 0, so we immediately ﬁnd that (

ρ(x, t) =

ρ∗

for

x < –vF t

0

for

x v vF t .

Thus, the question of how the solution will look within the hatched region of 31) M.J. Lighthill, C.B. Whitham: Proc. Roy. Soc. A 229 (1955) 317–345 31) Bernhard Riemann (1826–1866); Göttingen

39

40

2 Weak Solutions of Conservation Laws

Fig. 2.2, i.e., for –vF t < x < vF t,

t>0,

is open. It can easily be veriﬁed that (

ρ1 (x, t) :=

ρ∗

for

x<0

0

for

xv0

∀tv0

(2.11)

is a weak solution of „ „ «« ρ =0, ∂ t ρ + f v · ∂x ρ 1 – ∗ ρ i.e., it is of the Greenshields type of (2.3) together with the initial condition (2.10), because, if realized for our particular ﬂux f, the left hand side of (2.2) leads to „ «– Z∞ Z∞ » Z∞ ρ1 ∂t Φρ1 + ∂x ΦvF ρ1 1 – ∗ d(x, t) + Φ(x, 0)ρ0 (x) dx ρ 0 –∞

Z∞ Z0

=

–∞

ρ∗ ∂t Φ d(x, t) +

0 –∞

= ρ∗

8 Z0
–∞

:

0

Z0

Φ(x, 0)ρ∗ dx

–∞

∂t Φ dt + Φ(x, 0)

9 = ;

dx = ρ∗

Z0

{Φ(x, 0) – Φ(x, 0)} dx = 0 .

–∞

In other words, (2.1) is a solution which keeps its initial state. The drivers do not move even though the trafﬁc light has switched from red to green. This solution describes a special situation where the state along the left hand boundary of the hatched region of Fig. 2.2, i.e., along the characteristic t = – vxF , is connected with the right hand boundary t = vxF across a line of discontinuity.

Fig. 2.2 Characteristics for the dissolution of a trafﬁc jam.

2.2 Trafﬁc Flow Example with Loss of Uniqueness

Of course, drivers do not behave this way, so (2.11) is certainly an unreasonable and hence irrelevant solution. The drivers’ ride impulse has not yet been introduced into the model. However, there is another weak solution to our problem, namely 8 ρ∗ for x < –vF t , t v 0 > > > < v t – x ρ2 (x, t) := ρ∗ F (2.12) for –vF t u x u vF t , t > 0 , > 2vF t > > : 0 for x > vF t , t v 0 which can also easily be veriﬁed by elementary integration procedures. Within this solution, the states along the left and the right hand sides of the hatched region of Fig. 2.2 are continuously connected. In this case, the density decreases, so this type of a solution is called a rarefaction wave. The head of the trafﬁc jam, i.e., the the front end of the chain of cars at a standstill, withdraws against the direction of movement of the trafﬁc with speed –vF , whereas the cars in front move with speed +vF . Obviously, (2.12) coincides with our experiences within the bounds of Greenshields’ idealizations. It must therefore be regarded as the particular weak solution that ﬁts the circumstances of the given real situation. 33) Of course, loss of uniqueness can also occur if systems of conservation laws are involved instead of scalar situations. As another application of Greenshields’ trafﬁc ﬂow model, let us consider the propagation of the end of the trafﬁc jam opposite to the direction of the movement. We start from an initial situation at the instant t = 0, where the trafﬁc shows ∗ maximal ﬂux on the left of x = 0, i.e., ρ = ρ2 , whereas the cars on the right of x = 1 have already stopped, i.e., ρ = ρ∗ , and where we have a continuous linear transition in-between: 8 ∗ ρ > > for x < 0 > > > <2 ∗ ρ0 (x) = ρ (1 + x) (2.13) for 0 u x u 1 . > > 2 > > > : ∗ for x > 1 ρ The characteristics then turn out to be x = x0

„ « 1 x t= 1– vF x0 x – x0 t=– vF

for for for

9 > > > > > = 0 < x0 u 1 . > > > > > ; x0 > 1

x0 u 0

33) This is also an example of a situation mentioned earlier, namely that a solution that is continuous for t > 0 can develop from a discontinuous initial function.

(2.14)

41

42

2 Weak Solutions of Conservation Laws

vF

Fig. 2.3 Characteristics at the creation of a trafﬁc jam.

This solution ρ(x, t) maintains its continuity for the time 0 u t < v1F . Then, a discontinuity suddenly occurs at t = v1F , and this discontinuity moves for t > v1F along an as-yet unknown curve Γ. However, we know that this curve occurs in the upper left of the plane because the characteristics arising from the x-axis on the right of x0 = 1 undoubtedly intersect with the characteristics arising from x0 < 0. As soon as Γ becomes known, the values of ρ given along the characteristics make it possible to explicitly compute a solution ρ(x, t). It is discontinuous along Γ but still an element of Lloc 1 , and thus actually a weak solution. The drivers do not expect such a discontinuity to crop up, and this is often one of the reasons for a rear-end collision. One of the examples in the next section will show how Γ can be determined.

2.3 The Rankine–Hugoniot Condition

We are going to treat the particular case where a certain discontinuity of a weak solution V of (2.1) moves continuously along the x-axis. In other words, the set of points of Ω where these discontinuities occur is a continuous curve Γ in Ω. In this case, it may be that all of the components of V along this path are discontinuous, or that this is only true of some of them. In gas dynamics, often the density is particularly disturbed, resulting in shocks. Let P0 = (x0 , t0 ) ∈ Γ with t0 > 0, and assume that a bounded neighborhood D(P0 ) ⊂ Ω of P exists which does not contain points on the x-axis and where V is smooth outside Γ. 34) We decompose D(P0 ) into two parts, D1 and D2 (cf. Fig. 2.4), each of which we are going to treat as closed domains. Taking the smoothness of V in the open domains D1 and D2 into account, for a particular Φ ∈ C10 (Ω) (the compact support of which coincides with D(P0 ) = 34) The considerations to be formulated here can easily be generalized in order to ﬁt more complicated situations too.

2.3 The Rankine–Hugoniot Condition

0

Fig. 2.4 Sketch to explain the proof of the Rankine–Hugoniot condition.

D1 ∪ D2 ), we ﬁnd on D1 and on D2 the relation Z

–

ˆ

Z∞

˜

V∂t Φ + f (V )∂x Φ d(x, t) –

Ω

Z

=–

V0 (x)Φ(x, 0) dx

–∞

ˆ

˜ ∂t (VΦ) + ∂x ( f (V )Φ) d(x, t) +

D1

Z

–

Z

ˆ

˜ ∂t VΦ + ∂x f (V )Φ d(x, t)

ˆ

˜ ∂t Vt Φ + ∂x f (V )Φ d(x, t)

D1

ˆ

Z

˜

∂t (VΦ) + ∂x ( f (V )Φ) d(x, t) +

D2

D2

=0. Together with the validity of (2.1) in D1 and in D2 , Z Z ˆ ˜ ˆ ˜ ∂t (VΦ) + ∂x ( f (V )Φ) d(x, t) + ∂t (VΦ) + ∂x ( f (V )Φ) d(x, t) = 0 . D1

D2

If applied to each of the components of both integrals, the divergence theorem then leads to Z Z {(VΦ) dx – ( f (V )Φ) dt} + {(VΦ) dx – ( f (V )Φ) dt} = 0 . ∂D1

∂D2

Because Φ vanishes at the boundary of its support, i.e., at the boundary of D, we ﬁnd ZQ2 ZQ2 {(VL Φ) dx – ( f (VL )Φ) dt} – {(VR Φ) dx – ( f (VR )Φ) dt} = 0 , Q1

Q1

where VL denotes the left hand limits of V along Γ if the values of x approach Γ from the left for every ﬁxed value of t, i.e., VL (x, t) = V(x – 0, t) for (x, t) ∈ Γ. Analogously, VR denotes the values of V if we approach Γ from the right. The integrals should be considered line integrals along Γ.

43

44

2 Weak Solutions of Conservation Laws

Let [V ] := VL – VR ,

[ f ] := fL – fR := f (VL ) – f (VR )

be the jumps. Then, obviously, ZQ2 j

[V ] Q1

ﬀ dx – [ f ] Φ dt = 0 , dt

ˆ denotes the velocity with which the discontinuity moves along the where dx dt =: v x-axis. Since the test function Φ with support D(P0 ) is arbitrary, and because P0 is an arbitrarily chosen point on Γ, the so-called Rankine–Hugoniot condition [V ]ˆ v = [f]

(2.15)

is found to hold along Γ component-wise under the conditions that were assumed to hold. Equation (2.15) is also called the jump condition, and for every weak solution V there is a particular velocity vˆ = vˆ(V ). Example If (2.15) is applied to our one-dimensional gas ﬂow where 0 0 1 ρ B C V = @qA , E

q

1

B C B1 2 C B q + pC f = Bρ C, B C @ E+p A q ρ

(2.16)

and if q = ρu, with the ﬂow velocity u taken into account, we ﬁnd from the ﬁrst component of the jump condition (ρL – ρR )ˆ v = q L – q R = ρ L uL – ρ R uR i.e., ρL (ˆ v – uL ) = ρR (ˆ v – uR ) . The second component leads to (qL – qR )ˆ v=

q2L q2R – + pL – pR , ρL ρR

(2.17)

2.3 The Rankine–Hugoniot Condition

i.e., v – uL ) = ρR uR (ˆ v – uR ) + pL – pR , ρL uL (ˆ

(2.18)

and the third component gives (EL – ER )ˆ v = EL

qL qR qL qR – ER + pL – pR , ρL ρR ρL ρR

i.e., v – uL ) = ER (ˆ v – uR ) + pL uL – pR uR . EL (ˆ

(2.19)

uL = uR = vˆ ,

(2.20)

If

(2.17) is fulﬁlled, and (2.18) then gives pL = pR , such that (2.19) is also realized. Hence, in this case, the discontinuity “swims” with the ﬂow; this means that there are no gas particles that cross the point of discontinuity, though ρL = ρR or EL = ER do not necessarily hold. The situation described by q L ρR = q R ρL ,

ρL =/ ρR

(2.21)

is called contact discontinuity. Otherwise, uL =/ vˆ, uR =/ vˆ show that there are gas particles which cross the discontinuity, because in this case (2.17) shows that vˆ – uL and vˆ – uR are both positive or both negative. 35) This type of discontinuity is called a shock, and the curve Γ is also called a shock or a shock curve. The particular side of the shock where particles that have already crossed the shock are situated is called the back side of the shock, while the other part is called the front side. If, for instance, vˆ > uR > 0, the right hand side is the front side and the left hand side is the back side. Assume a weak solution to be smooth outside the discontinuity curve Γ, hence to be outside Γ also a solution of (2.1). The weak formulation (2.2) of the problem under consideration is then equivalent to the demand that the Rankine–Hugoniot condition is fulﬁlled along Γ and that the original version of the initial value problem (2.1) is simultaneously fulﬁlled outside Γ too. This equivalence follows from the already proven fact that weak solutions are also classical solutions in domains where these weak solutions are smooth. In the case of a linear system ∂t V + ∂x (AV) = 0 , the Rankine–Hugoniot condition leads to [V ] · vˆ = [AVL – AVR ] = A[V ] . 35) We omit the unnatural case ρ < 0.

45

46

2 Weak Solutions of Conservation Laws

Thus, vˆ is an eigenvalue of A as far as a discontinuity of the initial function occurs, i.e., as far as [V ] =/ 0 is found to hold for t = 0 (and thus also for increasing values of t). This leads toq x˙ = λk

for a particular k

since x(t) ˙ = vˆ is the differential equation for Γ. However, because of (1.18), this is also the differential equation for the k-th set of characteristic curves, so that discontinuities move along characteristics as far as linear problems are concerned. Let us wind up this section by determining the Γ in Greenshields’ trafﬁc ﬂow model for the case of a trafﬁc jam that increases in the direction opposite to the ﬂow direction. Because of the relation „ « ρ2 ρ2 fL – fR = vF ρL – L∗ – ρR + R∗ ρ ρ presented in the Greenshields model, and taking (2.15) into account, Γ has to be determined by the differential equation „ « ρL + ρR fL – fR = vF 1 – . x˙ = ρL – ρR ρ∗ From Fig. 2.3, the initial condition is „ « 1 =0. x vF

(2.22)

Moreover, Fig. 2.3 shows that ρ = ρ∗

for

t>

1 vF

along the characteristics from the right hand side, and this does not depend on the location of Γ. Along the vertical characteristics from the left side, we analogously ﬁnd that ρ=

ρ∗ 2

Hence ρL = x˙ = –

ρ∗ 2 ,

for

t>

1 . vF

and therefore

vF . 2

Together with (2.22), the shock is found to be the straight line t=

1 (1 – 2x) , vF

xu0.

2.3 The Rankine–Hugoniot Condition

vF

Fig. 2.5 Shock when a trafﬁc jam increases.

Knowing Γ and the values of ρ along the characteristic curves, we can now immediately construct a weak solution to our trafﬁc ﬂow example. In particular, the propagation of the end of the congestion against the direction of travel of the stillmoving cars can be evaluated: 8 ∗ ρ 1 > > for 0 u t < (1 – 2x) , –∞ < x u 0 > > 2 v > F < „ « x 1–x ρ(x, t) = ρ∗ 1+ for 0 u t < , 0ux<1 > > > 2 1 – vF t vF > > ∗ : ρ otherwise . (2.23) This solution shows that, under the assumptions of the Greenshields idealizations, and for the particular initial situation under consideration, that the end of the chain of stationary cars originally located at x = 1 will be located at x = 0 after v1F time units. From this moment on, the end of the trafﬁc jam moves with the congestion velocity 12 vF against the direction of the still-moving cars, and the transition from the density 12 vF of the moving trafﬁc to the density ρ∗ of the set of cars that have already come to a halt occurs very suddenly, at least from the driver’s point of view. With respect to the ﬁrst part of our Lighthill–Whitham–Greenshields trafﬁc ﬂow model, 36) as well as generally speaking, a question now arises: how do we characterize the particular solution that can be used to answer the real-world problem underlying the mathematical model for the non-unique weak solutions of (2.1)? We are going to address this question in the next chapter.

36) A more realistic but still continuous model can be found in Section 4.3.

47

3 Entropy Conditions 3.1 Entropy in the Case of an Ideal Fluid

The loss of uniqueness caused by adopting the concept of weak solutions leads to the need to formulate a criterion by which the physically relevant weak solution can be selected from the set of all of the weak solutions. We therefore require an additional constraint that will characterize this particular weak solution. In order to work out how to do this, let us look again at the important application of conservation laws in order to describe the dynamics of ideal gases. Besides the state equation, we took into account the conservation of mass, the conservation of momentum and the conservation of energy. The conservation of energy was formulated by the ﬁrst law of thermodynamics, but what we have not considered yet is the second law, and its assertion about the behavior of the state variable S = entropy/mass . Without going into detail, we will mention that the entropy describes a measure of the probability that a particular physical state exists (Boltzmann statistics in thermodynamics 37)). The second law of thermodynamics 38) can be stated as follows: If there is a closed physical system without any supply of energy from outside, every physical process inside the system takes a course such that the entropy does not decrease, 39) and it increases if possible. 40) Figure 3.1 shows an example of the validity of the second law, namely a shock tube divided by a membrane into two parts, one of which is ﬁlled with a gas while the other is evacuated. When the membrane is taken away, some of the gas will begin to move into the evacuated part of the tube, even though all of the conservation principles would also hold if the gas particles maintained their positions (in fact, the Euler equations would also hold in this case). However this is obviously not 37) Ludwig Boltzmann (1844–1906); Graz, Vienna, Leipzig, Vienna 38) R.J.E. Clausius (1822–1888); Zurich, Würzburg, Bonn

39) See, e.g., R. Ansorge, T. Sonar: ZAMM 77 (1997) 803–821 40) Irreversible processes

50

3 Entropy Conditions

Fig. 3.1 Example of increasing entropy.

a physically relevant solution, as everybody knows from experience. 41) The reason for this is that the equidistribution of the particles throughout the whole tube is a more probable state than the initial situation. In other words, we expect the second law to be a model for the formulation of a constraint that allows us to pick out the relevant solution from the set of weak solutions of a conservation law system, even when the problems are from other ﬁelds of application, such as economics. In any case, such a constraint will be called an entropy condition and the selected solution will be called an entropy solution; hopefully the only one obtained from a proof of uniqueness theorem. Generalizations of the second law in order to ﬁnd a constraint that works for more than just physical tasks have been derived by several authors, particularly Oleinik and Lax, who initially tackled this question in relation to scalar conservation laws. To understand their ideas, let us look at the situation in gas dynamics; for convenience, we consider a one-dimensional ﬂow. It was stated in Section 1.1 that each of the state variables can be uniquely expressed in terms of two of the other state variables via a so-called equation of state; (1.9) was a particular example. One result of the theory of thermodynamics leads to the relation dS =

˜ dε + p dW T

(3.1)

˜ denotes the speciﬁc volume W ˜ := 1 (volume of mass 1). where W ρ If we look for the connection between the three state variables ε, S and ρ = the equation of state belonging to these variables leads to

1 ˜, W

˜ , dε = ∂S ε dS + ∂W ˜ ε dW and hence with (3.1) to ˜ + p dW ˜ = ∂S ε dS + (∂W ˜ , T dS = ∂S ε dS + ∂W ˜ ε dW ˜ ε + p) dW such that ∂S ε = T ,

∂W ˜ ε = –p

˜ follows from the linear independence of dS and dW. 41) By the way, this is analogous to the situation where a trafﬁc light switches from red to green but the drivers do not move.

(3.2)

3.1 Entropy in the Case of an Ideal Fluid

Because of ˜ = T∂t S – p∂t W ˜ , ∂t ε = ∂S ε∂t S + ∂W ˜ ε∂t W the relation 9 ∂t ρ > > > ρ2 = ∂x ρ > > ; ∂x ε – p 2 .> ρ

˜ = ∂t ε – p T∂t S = ∂t ε + p∂t W follows. Analogously: T∂x S =

(3.3)

Denoting the entropy per volume as s := ρS, the quantity ∂t s + ∂x (us) can be expressed as ∂t s + ∂x (us) = ∂t (ρS) + ∂x ((ρu) · S) = ∂t ρS + ρ∂t S + ∂x (ρu)S + ρu∂x S = [∂t ρ + ∂x (ρu)] · S + ρ[∂t S + u∂x S] . The ﬁrst term on the right hand side vanishes because of the continuity equation, and taking (3.3) into account, the last equation leads to j„ « „ «ﬀ ρ ∂t ρ ∂x ρ ∂t s + ∂x (us) = ∂t ε – p 2 + u ∂x ε – p 2 . (3.4) T ρ ρ The conservation law (1.7) concerning the speciﬁc total energy E, and formulated for the particular situation of a one-dimensional ﬂow, reads „ « E+p ∂t E + ∂x q =0 ρ or, with E = ρε + ρ2 u2 (reformulated using primitive variables), j ﬀ 1 1 ∂t ρε + ρ∂t ε + ∂t ρu2 + ρu∂t u + u ∂x p + ∂x ρε + ρ∂x ε + ∂x ρu2 + ρu∂x u 2 2 “ ” ρ 2 + ρε + u + p ∂x u = 0 , 2 i.e., „ « „ « u2 p (∂t ρ + ∂x (ρu)) ε + + ρ ∂t ε + u∂x ε + ∂x u 2 ρ „ « ∂x p + ρu ∂t u + + u∂x u = 0 . ρ

(3.5)

Also, in this equation, the ﬁrst term vanishes because “ 2 ”of the continuity equation. q The second conservation law, namely ∂t q + ∂x ρ + p = 0, also rewritten using primitive variables, leads to „ « ∂x p ρ ∂t u + + u∂x u + u(∂t ρ + ∂x (ρu)) = 0 , ρ so that the third term on the left hand side of (3.5) also disappears, again through

51

52

3 Entropy Conditions

the use of the continuity equation. Thus, after dividing by ρ, ∂t ε + u∂x ε +

p ∂x u = 0 , ρ

(3.6)

remains, so the term inside the outer braces in (3.4) becomes p p ∂t ρ p u∂x ρ p – ∂x u – – = – 2 (∂t ρ + ∂x (ρu)) = 0 , ρ ρ ρ ρ ρ ρ

(3.7)

again because of the continuity equation. Hence, s also fulﬁlls a conservation law: ∂t s + ∂x (us) = 0 ,

(3.8)

and this holds automatically if the Euler equations are fulﬁlled. However, we have to take into account that this result requires that the differentiability properties we used to derive (3.8) are ensured. Thus, if the ﬂow is smooth, Section 1.1 shows that the total entropy Z s d(x, y, z) W(t)

of an volume W(t) of the ﬂuid selected arbitrarily at an arbitrary instant t stays constant with respect to time. This is known as isentropic ﬂow. Equation (3.8) cannot be expected to hold if the smoothness assumptions are not fulﬁlled. What can be expected, however, because of the second main theorem, is the validity of the inequality ∂t s + ∂x (us) v 0

(3.9)

in its weak form, i.e., Z Z (s∂t Φ + us∂x Φ) d(x, t) + s(x, 0)Φ(x, 0) dx u 0 Ω

∀ Φ ∈ C10 (Ω) with

R

(3.10)

Φv0.

Formally, and because it is an inequality, (3.9) was multiplied by nonnegative test functions, then integrated by parts, and ﬁnally this “derivation” of (3.10) should be forgotten. Equation (3.10) is the entropy condition for the ﬂow under consideration, and the “derivation” shows that it is indeed satisﬁed automatically provided that the original conservation laws are fulﬁlled according to the classical understanding. This follows from the fact that (3.8) was a direct consequence of the Euler equations. If the weak solution V of the Euler equations is discontinuous, and thus not necessarily unique, (3.10) is actually an additional condition.

s

Remark We mention without proof that the physical entropy per volume –s = –s(V ) in gas dynamics is strictly convex.

3.2 Generalization of the Entropy Condition

3.2 Generalization of the Entropy Condition

Lax 42) provided a deﬁnition of an entropy solution for a general system of conservation laws (2.1) that can, in the case of one spatial variable and by imitating the gas dynamics situation, be formulated as follows: Deﬁnition A weak solution V = (v1 , . . . , vm )T of the system (2.1), i.e., a solution of (2.2), is called an entropy solution of this system if there is a scalar and strictly convex func˜ = S(V ˜ ) as well as a scalar function F ˜ = F(V ˜ ) belonging to ˜ tion S S such that, in every domain where V is smooth, (2.1) leads automatically to the validity of ˜ ) + ∂x F(V ˜ )=0, ∂t S(V

(3.11)

and such that the entropy condition Z ˘ ¯ ˜ ˜ S(V(x, t))∂t Φ(x, t) + F(V(x, t))∂x Φ(x, t) d(x, t) Ω

Z

+

(3.12) ˜ 0 (x))Φ(x, 0) dx v 0 , S(V

∀ Φ ∈ C10 (Ω) , Φ v 0

R

is respected in the non-smooth situation.

s

Remarks ˜ is often also called the entropy functional and F ˜ is called the entropy ﬂux – S ˜ = –s – In gas dynamics we have S – On occasion we will talk more speciﬁcally of Lax entropy solutions, because some other deﬁnitions will also be given later – (3.12) is often abbreviated by the formulation: ˜ ) + ∂x F(V ˜ ) u 0 holds weakly ∂t S(V ˜ is already more or less determined by (3.11). Especially in the After choosing ˜ S, F scalar case m = 1(V = v, f = f ), (3.11) leads for smooth solutions v to ˜ (v) · ∂t v + F ˜ (v)∂x v = 0 , S and therefore with (2.1) to ˜ (v) {–∂x f (v)} + F ˜ (v)∂x v = 0 , S 42) P. Lax in: E. Zarantonello (ed.): Contributions to nonlinear functional analysis. New York: Academic 1971

53

54

3 Entropy Conditions

i.e., to ˆ

˜ ˜ (v)f (v) + F ˜ (v) ∂x v = 0 . –S

Because this must hold automatically, independent of the choice of vo (x), 43) the relation ˜ (v) = S ˜ (v)f (v) F must be fulﬁlled, i.e., ˜ = F(v)

Zv

˜ S (α)f (α) dα + const .

(3.13)

0

Only the constant remains arbitrary. If m > 1 and the function V is smooth, (3.11) yields ˙ ¸ ˙ ¸ ˜ ∂t V + ∇V F, ˜ ∂x V = 0 , ∇V S,

where ` ´T ∇V = ∂v1 , ∂v2 , . . . , ∂vm .

By analogy with m = 1, putting (2.1) into this equation, we ﬁnd ˙ ¸ ˙ ¸ ˜ J f (V )∂x V + ∇V F, ˜ ∂x V = 0 , i.e. – ∇V S, “ ” ˜ – (J f (V ))T ∇V S ˜ =0. (∂x V)T ∇V F

˜ and F ˜ have to be Again, because this relation must hold for any choice of V0 , S connected by ˜ – (J f (V ))T ∇V S ˜=0, ∇V F

(3.14)

˜ F ˜ (to be and this is a system of m differential equations for only two functions S; determined) that depends on the m independent variables v1 , . . . , vm . Hence, (3.14) does not necessarily have a solution when m > 2, 44) so the deﬁnition of an entropy condition used so far must be modiﬁed appropriately. We shall come back to this question in Section 3.3. However, if f (V ) is a gradient too, i.e., f (V ) = ∇V k(V ) 43) In particular, for initial functions that generate smooth solutions 44) As far as gas dynamics is concerned, ˜ = –s a suitable entropy ﬂux belonging to S exists, though 3 u m u 5.

3.2 Generalization of the Entropy Condition

˜ ) = 1 V, V turns (3.14) with a scalar function k(V ), the particular choice of S(V 2 into the following system of differential equations: ! m m X X ˜ ∂vi F = ∂vν vi kvν = ∂vi ∂vν kvν – ∂vi k (i = 1, . . . , m) , ν=1

ν=1

and this system can be satisﬁed by several solutions, e.g., by the particular entropy ﬂux ˜ ) = f (V ), V – k(V ) + const . F(V

(3.15)

A straightforward formulation of these considerations for m = 1 leads directly to the particular pair Zv ˜ = 1 v 2 , F(v) ˜ = f (α)α dα + const , S(v) (3.16) 2 0

but an inﬁnite number of other suitable pairs obviously also exist. This fact generates a further question, namely about the entropy solution’s inde˜ pendence of the choice of the entropy functional S. In order to answer this question, we cite the following theorem whose proof is more or less just a word-by-word copy of the proof of the Rankine–Hugoniot condition that takes (3.12) into account instead of (2.2): Theorem 3.1 ˜ of functions that belong to a weak solution V and Assume that there is a pair ˜ S, F fulﬁll condition (3.14); let the discontinuities of V form a curve Γ along which the Rankine–Hugoniot condition holds. Then it can be stated that V is a Lax entropy solution if and only if the inequality ˆ ˜ ˆ ˜ ˜ vˆ u F ˜ S

(3.17)

holds, where vˆ is the velocity that occurs in the Rankine–Hugoniot condition (2.15), i.e., the velocity with which the discontinuities move along the x-axis. ˆ ˜ ˆ ˜ ˜ := S(V ˜ := F(V ˜ L ) – S(V ˜ R ) and F ˜ L ) – F(V ˜ R ) denote the heights of As in (2.15), S the jumps. In the scalar case, and together with (2.15), (3.17) yields a particular form of the entropy condition equivalent to (3.12), namely ˆ ˜ ˜ [f] S

[v]

ˆ ˜ ˜ , u F

(3.18)

i.e., using (3.13), vL

Z ˜ L ) – S(v ˜ R) ` ´ S(v ˜ (α)f (α) dα u 0 . f (vL ) – f (vR ) – S vL – vR vR

45) As has been assumed to be the case so far

(3.19)

55

56

3 Entropy Conditions

This leads to the following theorem: Theorem 3.2 ˜ is strictly convex, 45) i.e., If the entropy functional S(v) ˜

(v) > 0 , S

∀v ∈ R ,

and if the ﬂux f is strictly convex, and also if discontinuities only occur along smooth curves Γ which do not intersect in the area under consideration, condition (3.19) only holds if and only if the jump relation vL v vR

(3.20)

is respected along each of these curves Γ. In particular, this result shows that all en˜ and their entropy ﬂuxtropy conditions created using strictly convex functionals S ˜ es F are equivalent, because these functionals do not play a role within Eq. (3.20). Proof: The function ˜f(v) deﬁned for every ﬁxed value of vr by 8 Zv > ˜ ˜ > < S(v) – S(vR ) ( f (v) – f (v )) – S ˜ (α)f (α) dα R ˜f (v) := v – vR > vR > : 0

for v =/ vR for v = vR

is differentiable, and its derivative reads ) j 8 ( ﬀ ˜ ˜ > <– S ˜ (v) – S(v) – S(vR ) · f (v) – f (v) – f (vR ) for v =/ vR ˜f (v) = v – vR v – vR . > : 0 for v = vR

The terms within the braces are positive for v > vR and negative for v < vR because ˜ S as well as f are strictly convex. This leads to ˜f (v) < 0

for

v =/ vR

so that vR is the only value for which ˜f vanishes, and ˜f decreases strictly monotonously for v =/ vR . Thus, ˜f (v) > 0 for v < vR ,

˜f (v) = 0 for v = vR ,

˜f(v) < 0 for v > vR .

(3.21)

Equation (3.19) implies that ˜f(vL ) u 0, and because of (3.21) this is really equivalent to vL v vR .

(3.22)

3.2 Generalization of the Entropy Condition

s

Remark The same arguments, if applied to our trafﬁc ﬂow example with its strictly concave ﬂux f (ρ) would similarly lead to the equivalence of the entropy condition to ρL u ρR ,46)

(3.23)

and the interpretation of this result shows that car drivers endeavor to smooth a discontinuous situation as quickly as possible, i.e., to reach an equals sign, 47)or that they aim to achieve a nondecreasing development of the trafﬁc density close to an actual discontinuity. In particular, car drivers do not stop when they notice a trafﬁc jam ahead; they would rather drive to the end of the jam so that the density increases there. When a driver stops, the end of the jam (i.e., the discontinuity) passes over him/her, moving against the direction that the cars are traveling in. The particular weak solution (2.11) of our trafﬁc ﬂow example obviously does not fulﬁll condition (3.23). In other words, (3.23) represents the driver’s ride impulse and completes the rough Lighthill–Whitham trafﬁc ﬂow model. 48)

There is also a geometric characterization of the entropy solution, provided that the ﬂux f is strictly convex: The Rankine–Hugoniot condition (2.15), together with the mean value theorem, yields

vˆ =

f (vL ) – f (vR ) = f (vR + ϑ(vL – vR )) , vL – vR

(0 < ϑ < 1) ,

and the strict convexity of f together with (3.20) then leads to f (vR ) < vˆ < f (vL ) .

(3.24)

Thus, if and only if the weak solution is the entropy solution, the slopes of the characteristics (1.17) belonging to this solution cause the characteristics to run into the shock curves Γ for increasing t instead of leaving them (cf. Fig. 3.2).

46) The strictly concave case can easily be transformed into a problem with a strictly convex ﬂux. 47) The discontinuity at the front of a trafﬁc jam is just such a situation. 48) R. Ansorge: Transpn. Res. 24B (1990) 133–143

57

58

3 Entropy Conditions

Fig. 3.2 Course of the characteristics for entropy solutions of scalar problems as soon as shocks appear.

As far back as 1957, Olga Oleinik published an entropy condition for weak solutions of scalar conservation laws. 49) It can be formulated as follows: Deﬁnition A weak solution v of the scalar problem (2.1) is called an entropy solution in the sense of Oleinik – or an Oleinik solution – if there is a constant E > 0 such that v(x + a, t) – v(x, t) E u , a t

∀a > 0 ,

∀t > 0 ,

∀x ∈ R .

(3.25)

Assume that the discontinuities of a weak solution of a one-dimensional scalar conservation law form a curve Γ such that the Rankine–Hugoniot condition (2.15) holds along this curve; if then the point (x, t) ∈ Ω (t > 0, ﬁxed) moves to Γ from the left, the point (x + a, t), a > 0, eventually becomes a point to the right of Γ such that (3.25) states v(x + a, t) – vL u

E a. t

If a now tends to zero, vR – vL u 0 follows.

s

Remark Thus, an Oleinik solution is a Lax solution, too. If – hopefully – a uniqueness theorem for the Lax entropy solution is available, the Oleinik solution is also uniquely determined and equals the Lax solution. This uniqueness problem for one-dimensional scalar conservation laws will be discussed in the next section, and the question concerning a suitable generalization of the deﬁnition of a Lax entropy solution to systems of conservation laws will then also be addressed. 49) O. Oleinik: Usp. Math. Nauk. (N.S.) 12 (1957) 3–73

3.3 Uniqueness of Entropy Solutions

s

Remark Another approach to the idea of an entropy solution for scalar problems comes from the well-known uniqueness of the solution vε (x, t) of the parabolic problem ∂t v + ∂x f (v) = ε∂xx v ,

ε>0

v(x, 0) = v0 (x) , if the right hand side of (2.1) is enriched – analogously to the linear problem (1.24) – by a diffusion term where the diffusion parameter ε tends to 0. 50)

3.3 Uniqueness of Entropy Solutions

In this section we are going to investigate the uniqueness of the Lax entropy solution. Lax himself gave the outline of the following theorem: Theorem 3.3 Let v˜ and v be two Lax entropy solutions of the scalar form of the conservation law (2.1) which are piecewise continuous for every ﬁxed t v 0, non-smooth only along certain curves Γ, and have initial functions v˜0 and v0 , respectively. The ﬂux f is assumed to be strictly convex. Then, the distance ˜ v (·, t) – v(·, t)L1 (I) monotonously decreases with respect to t, where the L1 -norm is understood to be taken over an arbitrary but ﬁxed interval I of the x-axis. 51) Proof: Decompose, for every ﬁxed t, the interval I into intervals (xν , xν+1 ) (ν = 0, ±1, ±2, . . .) in such way that v˜(x, t) – v(x, t) has the sign (–1)ν uniformly on (xν , xν+1 ). A decomposition of this type exists because v˜ – v is assumed to be piecewise continuous with respect to x. Evidently, xν depends on t: xν = xν (t) (ν = 0, ±1, ±2, . . .). Thus, ˜ v (·, t) – v(·, t)L1 (I) =

X ν

(–1)

ν

xZ ν+1 (t)

ˆ

˜ v˜(x, t) – v(x, t) dx .

(3.26)

xν (t)

50) N.N. Kuznetsov: USSR Comput. Math. Math. Phys. 16 (1976) 105–119 51) In particular, if then v˜0 = v0 in L1 , i.e., ˜ v (·, 0) – v(·, 0)L1 (I) = 0 for every interval I, ˜ v (·, t) – v(·, t)L1 (I) = 0 follows for all t v 0,

i.e., v˜ = v in L1 at every time level t, and this means uniqueness.

59

60

3 Entropy Conditions

v˜ and v are piecewise smooth on (xν (t), xν+1 (t)), and differentiation with respect to t yields 8 > <ˆ X ˜ d (–1)ν v˜(xν+1 (t), t) – v(xν+1 (t), t) x ν+1 (t) ˜ v (·, t) – v(·, t)L1 (I) = > dt : ν ˆ ˜ – v˜(xν (t), t) – v(xν (t), t) x ν (t) 9 xZ ν+1 (t) > = ˆ ˜ ∂t v˜ – ∂t v(x, t) dx . + > ; xν (t)

Using (2.1), xZ ν+1 (t)

xZ ν+1 (t)

∂t v(x, t) dx = – xν (t)

∂x f (v(x, t)) dx xν (t)

= – f (v(xν+1 (t), t)) + f (v(xν (t), t)) follows; analogously for v˜. Hence, j X ˜ ˆ d (–1)ν v˜(xν +,1 (t), t) – v(xν+1 (t), t) x ν+1 (t) (3.27) ˜ v (·, t) – v(·, t)L1 (I) = dt ν ˆ ˜ – v˜(xν (t), t) – v(xν (t), t) x ν (t) ˆ ˜ – f (˜ v (xν+1 (t), t)) – f (v(xν+1 (t), t)) ﬀ ˆ ˜ + f (˜ v (xν (t), t)) – f (v(xν (t), t)) .

We have to distinguish between two different situations: i) The function v˜(·, t0 ) – v(·, t0 ) is continuous at xν (t0 ). 52) This function then vanishes at xν (t0 ), i.e.: v˜(xν (t0 ), t0 ) = v(xν (t0 ), t0 ) .

(3.28)

ii) One of the functions v˜(x, t0 ), v(x, t0 ) is discontinuous at xν (t0 ). We therefore need to sum up the right hand side of (3.27) only for the particular numbers ν such that (xν (t), t) or (xν+1 (t), t) lie on curves of discontinuities of v˜ or v.

52) For the arbitrary ﬁxed value of t0 under consideration

3.3 Uniqueness of Entropy Solutions

Let us ﬁrst discuss the case where (xν+1 (t), t) lies on a curve of discontinuities of v˜, whereas v is smooth at this position. The end-point xν+1 of the interval (xν (t), xν+1 (t)) must then be looked upon as xν+1 – 0, i.e., v˜(xν+1 (t), t) = v˜ . If, for example, v˜ – v > 0 on (xν , xν+1 ), and thus v˜ – v < 0 on (xν+1 , xν+2 ), then ν is even and v˜L > vL = vR > v˜R

(3.29)

because xν+1 on (xν+1 , xν+2 ) has to be viewed as xν+1 + 0. This leads to j ˆ ˜ (–1)ν v˜(xν+1 (t), t) – v(xν+1 (t), t) x ν+1 (t) ﬀ ˆ ˜ v (xν+1 (t), t)) – f (v(xν+1 (t), t)) – f (˜ = (˜ vL – vL )x ν+1 (t) – f (uL ) + f (vL ) . Because of the smoothness of v at (xν+1 (t), t), x = xν+1 (t) can only be a point on the shock curve of v˜. The Rankine–Hugoniot condition (2.15) therefore makes it possible to estimate the right hand side of the last equation by (˜ vL – vL )

vR ) f (˜ vL ) – f (˜ – f (˜ vL ) + f (vL ) v˜L – v˜R » – f (˜ vL ) – f (˜ vR ) f (˜ vL ) – f (vL ) – u0, =(˜ vL – vL ) v˜L – v˜R v˜L – vL

which takes into account the fact that (3.29) yields v˜L – vL > 0, but that vR ) f (˜ vL ) – f (vL ) f (˜ vL ) – f (˜ – u0. v˜L – v˜R v˜L – vL Here, the last inequality results from v˜R < vL < v˜L , which also follows from (3.29), as well as from the strict convexity of f (cf. Fig. 3.3).

~v R Fig. 3.3 Sketch relating to Lax’ uniqueness theorem.

vL

~v L

v

61

62

3 Entropy Conditions

A similar result follows if (xν+1 (t), t) is a point on a curve which is a set of discontinuities of v˜ as well as v; in this situation, the inequality v˜L > vL > vR > v˜R must be taken into account instead of (3.29), and it is to this part of the proof that the jump condition vL > vR applies. The same reﬂections work for the values of v˜ and v at the position (xν (t), t), namely the disappearance or non-positivity of the terms. Here, xν (t) = xR must be observed in the case of a discontinuity. The case v˜ < v on (xν , xν+1 ) can be treated in the same way. Thus, d ˜ v (·, t) – v(·, t)L1 (I) u 0 dt follows. As already emphasized, for systems of conservation laws with more than two ˜ F) ˜ of an entropy functional S ˜ and an entropy ﬂux F ˜ connectequations, pairs (S, ed by (3.14) and suitable for distinguishing an entropy solution from other weak solutions do not necessarily exist. We continue to treat only one-dimensional problems, but we try to formulate a suitable analog to (3.24) for problems where the convexity of the ﬂux can also be suitably generalized. Let N ⊂ Rm be a subset of the m-dimensional Euclidian space, and assume that system (2.1) is strictly hyperbolic for all V = (v1 , v2 , . . . , vm )T ∈ N, i.e., the eigenvalues λi (V) of the Jacobian J f (V ) are real and different from each other for all V ∈ N. Moreover, let f (V ) ∈ C2 (N) and λ1 (V ) < λ2 (V ) < . . . < λm (V ) .

(3.30)

The eigenvalues maintain this order for all V ∈ N; otherwise a vector V ∈ N with λi (V ) = λk (V )

for a pair (i, k)

with i =/ k

would have to exist because of continuity reasons. However, this would contradict the strict hyperbolicity on N. The eigenvectors si (V ) (i = 1, 2, . . . , m) belonging to the eigenvalues λi (V ) (i = 1, 2, . . . , m), respectively, are linearly independent. Let V(x, t) be a solution of (2.1) which is smooth on the domain G ⊂ Ω and has V(x, t) ∈ N ,

∀ (x, t) ∈ G .

The i-characteristics x = x(i) (t) corresponding to this solution (cf. (1.18)) are uniquely determined from the Picard–Lindeløf theorem if the functions λi (V(x, t)) fulﬁll a Lipschitz condition with respect to x (i = 1, . . . , m) and provided initial states x(i) (t0 ) with (x(i) (t0 ), t0 ) ∈ G are prescribed. Of course, existence and uniqueness theorems with weaker assumptions are also available, e.g., the existence theorem of Peano combined with the uniqueness condition of Nagumo.

3.3 Uniqueness of Entropy Solutions

Then, along an i-characteristic, ˆ ˜ d (3.31) V(x(i) (t), t) = – J f (V(x(i) (t), t)) – λi (V(x(i) (t), t)) · I ∂x V(x(i) (t), t) , dt where I in this context means the m ~ m unit matrix. In the case of m = 1, the right hand side of (3.31) vanishes, as observed earlier, so that V = v is constant along a characteristic which therefore becomes a straight line. Does an analog exist for m > 1? In other words, do a weak solution V and a set of i-characteristics x(i) belonging to this solution exist such that the solution is constant along these characteristics, which therefore become straight lines? In the case of ∂x V(x(i) (t), t) =/ 0, a positive answer requires that the following relation is valid:

∂x V(x(i) (t), t) = si (V(x(i) (t), t)) ∀ t > 0 . 53) Solutions of this type can actually be speciﬁed. If we introduce so-called centered waves V = V ∗ , which are centered at a position P0 = (x0 , t0 ) ∈ G, i.e., waves of the form „ « x – x0 (3.32) V(x, t) = V ∗ t – t0 with ˆ ˜m V ∗ (ξ) ∈ N ∩ C1 (R)

∀ P = (x, t) ∈ G ,

ξ :=

x – x0 . t – t0

These centered waves solve (2.1) if they fulﬁll ` ´ ∂t V + ∂x f (V ) = ∂x ξJ f (V ∗ ) + ∂t ξI V ∗ (ξ) = 0 which can also be described as (J f (V ∗ (ξ)) – ξI) V ∗ (ξ) = 0 . Hence, for one i ∈ (1, 2, . . . , m), the pair of relations ` ´ V ∗ (ξ) = si V ∗ (ξ) , ` ´ (for all possible values of ξ) ξ = λi V ∗ (ξ)

(3.33)

(3.34)

must hold. In this case, differentiation of the second equation of (3.34) with respect to ξ together with the ﬁrst equation yields (cf. footnote 51) 1 = ∇V λi (V ), si (V ) .

(3.35)

Deﬁnition If ∇V λi (V ), si (V ) = 0 holds for all V ∈ N with a set N ⊂ Rm , the i-th characteristic ` ´ ﬁeld λi (V ), si (V ) is termed genuinely nonlinear on N. 53) Here, a suitable norm of the vectorﬁeld V is chosen.

63

64

s

3 Entropy Conditions

Remark Because the scalar case implies λ(V ) = λ(v) = f (v), which yields ∇V λ(V ) = f

(v) , (3.35) can only be fulﬁlled if f

(v) =/ 0, and s(v) in this case is a nonzero scalar factor s(v). It can be normalized by s(v) = |f

1(v)| , which makes (3.35) equivalent to f

(v) > 0, i.e., to strict convexity of the ﬂux (or, if we normalize by concavity).

–1 |f

(v)| ,

to strict

We are therefore going to consider (3.35) as the transition announced earlier of the strict convexity of the ﬂux as deﬁned for scalar problems to the i-th characteristic ﬁeld in the case of m > 1. Let us now assume that N is a simply connected domain in Rm . Moreover, let ` ´ ` ´ V 0 ∈ N V 0 =/ 0 be a given state and ξ0 := λi V 0 . If ξ0 is used as starting point with an initial condition V ∗ (ξ0 ) = V 0 ,

(3.36)

the second equation in (3.34) will obviously be fulﬁlled at this point ξ0 . The ﬁrst equation of (3.34) represents a system of ordinary differential equations where (3.36) is used as the initial condition. We know that there is a unique solution for this system in a certain neighborhood of (ξ0 , V 0 ) provided that the assumptions of a standard theorem like the Picard–Lindeløf theorem are fulﬁlled. This solution runs with respect to V in N; let’s say for ξ0 – a u ξ u ξ0 + a with sufﬁciently small a > 0. Also, (3.35) then holds for V = V ∗ (ξ) because of the assumptions. Since the ﬁrst equation of (3.34) was also fulﬁlled, (3.33) also holds. Integration of (3.33) together with (3.36) then yields the fact that the second equation of (3.34) is also fulﬁlled, at least for the values of ξ considered here and if a can be diminished somewhat if necessary. We conclude that there is a nonvanishing smooth solution V, namely a wave centered at P0 , which belongs to a given nonvanishing smooth initial state V 0 and to a particular integer i, and leads to an i-th characteristic ﬁeld which is genuinely nonlinear on N: „ « ` ` ´´ x – x0 V(x, t) = V ∗ , V ∗ λi V 0 = V 0 (3.37) t – t0 for all (x, t) ∈ Ω with ` ´ ` ´ x – x0 u λi V 0 + a where a > 0 is sufﬁciently small . λi V 0 – a u t – t0 (3.38)

Along each of the straight lines x – x0 = c = const t – t0

3.3 Uniqueness of Entropy Solutions

with ` ´ ` ´ λi V 0 – a u c u λi V 0 + a

forming a fan, the solution „ « x – x0 ∗ = V ∗ (c) V t – t0 is constant.

Fig. 3.4 Fan of i-characteristics of centered waves.

Thus, along these straight lines given by x(i) (t) = x0 + c(t – t0 ) , the relation d d V(x(i) (t), t) = V ∗ (c) = 0 dt dt

(3.39)

holds. However, because V ∗ (c) is an eigenvector belonging to λi (V ∗ (c)), this also applies to ∂x V(x(i) (t), t) =

1 V ∗ (c) . t – t0

It follows from (1.18) and (3.33) that the straight lines that generate the fan and belong to this particular integer i are the i-characteristics formed using solution (3.32) through the point P0 , and the centered waves constructed so far are constant along these lines. Let us now assume that V 1 is a state in the neighborhood of V 0 . Then, by conti` ´ ` ´ nuity arguments, λi V 1 lies in a neighborhood of λi V 0 such that ˛ ` 1´ ` ´˛ ˛λi V – λi V 0 ˛ u a can be fulﬁlled for all states V 1 sufﬁciently close to V 0 .

65

66

3 Entropy Conditions

In particular, if ` ´ ` ´ ` ´ λi V 0 < λi V 1 u λi V 0 + a ,54)

we obtain the result „ « x – x0 ∗ V = V0 t – t0 along the straight lines t – t0 =

1 ` ´ (x – x0 ) , λi V 0

so the solution V ∗ (ξ) is constant along the lines and equals V 0 if this is its value at ξ0 . The same arguments show that the solution equals V 1 along the straight lines t – t0 =

1 ` ´ (x – x0 ) λi V 1

if this is its value at ξ0 . ` ´ ` ´ In other words, in the case of λi V 0 < λi V 1 , the state V 0 can be smoothly connected from the right with the state V 1 by a centered i-rarefaction wave. 55)

s

Remark ` ´ ` ´ ` ´ In`the´ scalar case, and if f is strictly convex, λi V 0 < λi V 1 means f v 0 < f v 1 , and so

v0 < v1 . In other words, if t is ﬁxed, the states decrease for decreasing x, i.e., from the right to the left (rarefaction). 56) Indeed, if the entropy condition (3.20) is fulﬁlled, this rarefaction can only be caused by a continuous process, because vL = vR . Assume now that system (2.1) has discontinuous solutions as well as or instead of smooth rarefaction waves, and let us assume that these discontinuities form certain curves Γ along which the Rankine–Hugoniot condition is satisﬁed. We again require conditions that are easy to apply in order to uniquely characterize the physically relevant entropy solution. 57) Here, we restrict ourselves to problems whose characteristic ﬁelds are genuinely nonlinear on N = Rm . 58) 54) In the case of λi (V 1 ) < λi (V 0 ), swap the roles of V 1 and V 0 . 55) See also the deﬁnition that follows Eq. (3.43) 56) If the ﬂux f is a concave function, like the ﬂux in the trafﬁc ﬂow model, the word “left” should be replaced by “right” and “decrease” by “increase.”

57) Recall that condition (3.12) cannot necessarily be applied when there are more than two equations. 58) This is analogous to scalar problems, i.e., to m = 1 where only convex ﬂuxes are taken into account.

3.3 Uniqueness of Entropy Solutions

A weak solution will be called an entropy solution if it respects a condition of the following type: – If initial values are prescribed along a non-characteristic curve, the condition will only be fulﬁlled by one weak solution – The condition coincides with (3.24) for scalar problems If [V ] = VL – VR are the jumps along a curve Γ of discontinuities of a weak solution moving along the x-axis at a speed vˆ, let k be the particular integer with λk (VR ) < vˆ < λk+1 (VR ) ,

k ∈ {1, 2, . . . , m – 1} ,

(3.40)

or vˆ < λ1 (VR ) or vˆ > λm (VR ). 59) Only the values of V0 along the particular characteristics through P0 which show the steeper ascents 1 , λk (V0 )

1 1 , ... , λk–1 (V0 ) λ1 (V0 )

(cf. Fig. 3.5)

are available in order to determine the m components of VR at P0 ∈ Γ from the given initial values.

Fig. 3.5 Sketch concerning the completeness of the information 60) needed in order to ensure uniqueness.

If, for example, a system of decoupled equations is under consideration, 61) one and only one of the m components of VR (P0 ) can be determined along each of these k steeper characteristics. Hence, m – k pieces of information are missing. If, at the same time, λj (VL ) < vˆ < λj+1 (VL ) , holds, corresponding arguments lead to the result that there are only m – j pieces of information available for the computation of the m components of VL . Hence, j pieces of information are missing. 59) See (3.30) 60) The particular choice of 0 ∈ Γ in Fig. 3.5 does not lead to a loss of generality. 61) See the linear situation (1.19)

67

68

3 Entropy Conditions

Thus, m – k + j pieces of information are missing for the unique determination of the two quantities VL and VR . However, m additional pieces of information concerning the relations between VL , VR and vˆ follow from the Rankine–Hugoniot condition (2.15). One of them must be used for the elimination of vˆ. Thus, the real number of pieces of missing information is only 1– k+ j. In other words, uniqueness can only be expected for k=j+1, i.e., if the relations λk (VR ) < vˆ < λk+1 (VR ) ,

λk–1 (VL ) < vˆ < λk (VL )

(3.41)

hold simultaneously. Although this result was only illustrated by an example, we accept that it provides a hint about a general deﬁnition of an entropy solution: Deﬁnition V is called an entropy solution if there is an index k for which (3.41) holds. A discontinuity of this type is then called a k-shock and the inequalities (3.41) are called entropy inequalities or (Lax) shock conditions. Equation (3.41) includes λk (VR ) < vˆ < λk (VL ) , so that (3.41) really coincides with (3.24) as far as scalar problems are concerned. Thus, both of the properties claimed earlier with respect to the formulation of an entropy condition for weak solutions of systems of conservation laws are respected. There are some connections between an i-characteristic ﬁeld and the so-called i-Riemann invariant: Deﬁnition A scalar function w = w(V ) deﬁned for all V ∈ N ⊂ Rm is called an i-Riemann invariant on N if ∇V w(V ), si (V ) = 0 ∀ V ∈ N .

s

(3.42)

Remark Thus, the property that an i-characteristic ﬁeld is genuinely nonlinear coincides with the property that λi (V ) is not an i-Riemann invariant. If the graph of a centered i-rarefaction wave V is situated in N, its i-Riemann invariants are constant. This follows from (3.34): ` ´ ∂x w V(x, t) = ∇V w(V ), si (V )V=V ∗ (x, t) = 0 (3.43) ` ´ and similarly ∂t w V(x, t) = 0.

3.4 Kruzkov’s Ansatz

Deﬁnition If a solution V of system (2.1) belonging to any initial function V0 is smooth on a region G ⊂ Ω, and if all the i-Riemann invariants w(V(x, t)) are constant on G, V is called an i-simple wave or an i-rarefaction wave on G.

3.4 Kruzkov’s Ansatz

The approach used by S.N. Kruzkov 62) to introduce the idea of a weak solution of problem (2.1) seems to differ from that of Lax. In order to understand the situation, it is sufﬁcient to restrict the presentation of Kruzkov’s ansatz to one-dimensional scalar problems. Kruzkov tries to ﬁnd his weak solutions in the space Lloc 1 (Ω), as already done in the Lax deﬁnition of a weak solution. Moreover, the test functions used in (2.2) by Lax also play a crucial role in Kruzkov’s deﬁnition, with the one restriction that these functions now vanish along the whole boundary of their compact support, i.e., also along parts of the boundary belonging to the x-axis: Φ(x, 0) = 0. This leads to the need to pay attention to the initial function in a different way than by the second integral in Eq. (2.2). Kruzkov calls the function v ∈ Lloc 1 (Ω) a weak solution of the problem ∂t v + ∂x f (v) = 0 ,

v(x, 0) = v0 (x)

if the following two conditions are fulﬁlled: Z j ∂t Φ(x, t)|v(x, t) – c| + ∂x Φ(x, t) Ω

(3.44)

ﬀ sgn (v(x, t) – c) [f (v(x, t)) – f (c)] d(x, t) v 0

∀ Φ ∈ C10 (Ω) with Φ v 0 and with Φ(x, 0) = 0 ,

∀c ∈ R ,

and ZR

|v(x, t) – v0 (x)| dx = 0

lim t→0

∀R ∈ R

(t ∈ [0, T]\˜ε)

(3.45)

–R

for T > 0 and for a set ˜ε of measure zero so that v(x, t) is well deﬁned almost anywhere as a function of x for any ﬁxed t ∈ [0, T]\˜ε. Kruzkov was able to show that there is a unique weak solution in Lloc 1 (Ω) in the sense of (3.44), (3.45) provided that the ﬂux f is smooth and strictly convex. Smoothness and strict convexity of the ﬂux are assumptions that were also made by Oleinik and Lax. 62) S.N. Kruzkov: Soviet Math. Dokl. 10 (1969) 785–788

69

70

3 Entropy Conditions

Hence, in Kruzkov’s deﬁnition, the characterization of an entropy solution does not consist of an Eq. (2.2) to be solved for all test functions together with an additional entropy condition (3.12), but simply an inequality. 63) One can say that Kruzkov’s deﬁnition already includes an entropy condition. Indeed, as far as the scalar problem is concerned, the left side of (3.44) corresponds with (3.12) if Φ(x, 0) = 0 is taken into account, and if for every particular ˜ is identiﬁed with |v – c| and the entropy constant c ∈ R the entropy functional S(v) ˆ ˜ ˜ ﬂux F(v) with sgn (v – c) f (v) – f (c) . ˜ and F ˜ are chosen in this way, the connection between these two functions If S demanded by (3.13) is also kept piecewise, namely for all values of v with the ex˜ is also convex – as entropy ception of v = c. Moreover, the particular functional S functionals are expected to be – but unfortunately no longer strictly convex. On the other hand, there is a somewhat stronger demand than formulated by Lax, namely that the validity of (3.44) is ensured for not only all nonnegative test functions Φ but also all c ∈ R. Thus, (3.45) must now hold for all test elements ˘ ¯ ˆ ∈ (Φ, c)|Φ ∈ C10 (Ω), Φ v 0, Φ(x, 0) = 0; c ∈ R . Φ

(3.46)

It can be shown by more than one chain of evidence that Kruzkov’s weak solution coincides with the Lax–Oleinik entropy solution provided that the ﬂux is smooth and strictly convex. One of these chains follows from numerical aspects. There are ﬁnite difference approximations whose numerical solutions converge for decreasing step sizes to the Lax–Oleinik entropy solution, but also (as can be proved by a slightly different convergence argument) to the Kruzkov solution. 64) It should be noted that problems can arise from ﬁelds of applications where the assumptions of smoothness and strict convexity of the ﬂux are violated. However, Kruzkov, together with Panov, showed that a unique solution for the problem associated with (3.44) and (3.45) does exist, even if f is merely continuous. 65) Thus, in the scalar situation, the Kruzkov concept of an entropy solution is more far-reaching. However, one has to pay for the renunciation of convexity and smoothness. It can happen that one of the properties of the entropy solution vE that is important from the point of view of numerical methods gets lost; namely, that an initial function v0 with compact support leads to a compact support of vE (·, t) (for every ﬁxed t > 0), as is the case if f is smooth and convex. A test example of this loss is given by |v|α with 0 < α < 1 , α 8 > x < –1 <0 for v0 (x) = 1 for –1 u x u 0 . > :0 for x>0 f (v) =

63) As well as a condition that respects the initial condition in a suitable way 64) A non-numerical argument concerning the connection between both deﬁnitions can be found in the paper mentioned in footnote 2.

(3.47)

65) S.N. Kruzkov, E.Y. Panov: Soviet Math. Dokl. 42 (1991) 316–321

3.4 Kruzkov’s Ansatz

vE

vE

vE

Fig. 3.6 Kruzkov–Panov solution (3.48) for 0 u t u

The solution reads 8 > 0 > > > < 1 vE (x, t) = „ « 1 > > t 1–α > > : x

α . 1–α

for for

t > α(x + 1) x < t u α(x + 1)

for

tux

(3.48)

(cf. Fig. 3.6). Whereas the smooth and strictly convex case behaves hyperbolically, a merely continuous ﬂux can lead to a problem with more parabolic character. Numerical procedures for problems of this particular type were recently developed under suitable conditions for the ﬁrst time by M. Breuss. 66) Here, the Kruzkov–Panov example (3.47) already shows that the important CFL condition (cf. Sections 6.1, 7.3) cannot be fulﬁlled when explicit ﬁnite difference methods (cf. Sections 4.1, 6.3, 7.2) are used. Thus, implicit methods must be taken into account, and this can actually be done successfully. Examples of mathematical models of real-world problems in ﬂuid mechanics, where standard assumptions like strict hyperbolicity, strict convexity or smoothness of ﬂux are not necessarily fulﬁlled, occur in the theory of ﬂows through porous media, e.g., in relation to oil production. The scalar so-called nonstandard Buckley–Leverett equation „ « v2 ∂t v + ∂x =0, a∈R v 2 + a(1 – v)2 models the displacement of an oleic phase by an aqueous phase in a porous medium. In this situation, the ﬂux is neither convex nor concave. Tveito and Winther 67) considered the following system theoretically with respect to stability: ∂t s + ∂x f (s, c) = 0 ` ´ ` ´ ∂t sc + a(c) + ∂x cf (s, c) = 0 . 66) M. Breuss: Numerik von Erhaltungsgleichungen in Nicht-Standard-Situationen. Ph.D. Thesis, Dept. of Mathematics, University of Hamburg 2001

67) In: B. Engquist, B. Gustafsson (eds.): Proceedings of the Third International Conference on Hyperbolic Problems. Uppsala: Chartwell-Bratt 1991, pp. 888–898

71

72

3 Entropy Conditions

This system also models the displacement of oil by water in a porous medium. However, in this case, the water is thickened by dissolved polymer in order to increase its viscosity. The difﬁculty of this example arises from the fact that the eigenvalues of the Jacobian are real but can coincide, so the system is hyperbolic but not necessarily strictly hyperbolic.

73

4 The Riemann Problem 4.1 Numerical Importance of the Riemann Problem

When we treated the trafﬁc jam dissolution example in Section 2.2 we met the ﬁrst time with a Riemann problem. It consisted of a conservation law of type (2.1) together with an initial function that was piecewise constant with certain jumps at isolated positions. One tool that can be used to solve differential equations numerically is the socalled ﬁnite difference method (FDM), where one tries to approximate the unknown values of the exact solution at isolated points (see also Chapters 6 and 7). (n) Particularly in the case of problem (2.1), we would like numerical values Vν at isolated positions (xν , tn ) on the upper half of plane Ω that are expected to be good approximations to the values VE (xν , tn ) of the exact entropy solution at these points. One particular approach to constructing FDMs involves, in a ﬁrst step, covering the positive time axis with a not necessarily equidistant grid {tn |(n = 0, 1, . . .)} of time step sizes Δtn := tn+1 – tn and ﬁxing isolated grid points xν (ν = 0, ±1, ±2, . . .) along the x-axis. Thus, by generating the Cartesian product of these two sets, Ω will be covered with a net of grid points parallel to the axes. Also, the step sizes hν := xν+1 – xν do not necessarily need to be equidistant (cf. Fig. 4.1).

Fig. 4.1 Grid of an FDM.

74

4 The Riemann Problem (n+1)

If the values Vν at time step t = tn+1 are only computed based on knowledge (n) of the values Vν that were computed at the previous time step, we call the scheme a one-step method. In contrast, multistep methods include approximate values for earlier steps, but we are going to restrict the present survey to one-step methods. Let t0 = 0 be the initial level, with approximate values

(0)

Vν

2 := xν+1 – xν–1

1 2 (xν +xν+1 )

Z

V0 (x) dx

(ν = 0, ±1, ±2, . . .) .

(4.1)

1 2 (xν–1 +xν )

It seems reasonable to describe the difference between the approximate solution and the unknown exact solution in terms of the topology of the space in which the original problem is based, which is the space Lloc 1 (Ω) in this particular case. This difference or truncation error should be estimated using a suitable norm. Hopefully, these errors will decrease for decreasing step sizes and will tend to zero sufﬁciently fast if the step sizes tend to zero. If this holds, the method is said to be convergent. Obviously, a direct comparison of the approximate solution with the exact solution n in o terms of the toplology of the original space can happen if the discrete values (n) Vν of the approximate solution deﬁned at isolated points can be appropriately extended to the intergrid points, e.g., by ˆ t) := Vν(n) V(x,

for

xν–1 + xν xν + xν+1 <xu , 2 2

tn u t < tn+1

(4.2)

(ν = 0, ±1, ±2, . . . ; n = 0, 1, 2, . . .) . Using this special extension or reconstruction, the discrete function now becomes ˆ that is constant on each of the described rectangles. Hence, V ˆ is a step a function V function on Ω and, thus, an element of Lloc (Ω). 1 ˆ to the time step t = tn , this restricted function V(x, ˆ tn ) of the space If we restrict V variable x is a step function, too, and from this, the one-step method together with ˆ tn+1 ). the reconstruction now generate V(x, ˆ tn ) is used as an initial function for the original problem in order to genIf V(x, erate a weak solution of the differential equation in (2.1) for the region t v tn , a Riemann problem arises. Its exact solution, if restricted to the time step t = tn+1 , would lead to values at the grid points at this step that could be considered approximations to the weak solution of the original problem at these new grid points. This idea was ﬁrst used by S.K. Godunov, 68) and has since been used by various other authors as the basis for further numerical schemes. This enabled effective approximate Riemann solvers to be developed, including local linearizations. We are therefore now going to study the Riemann problem, at least for linear problems with constant coefﬁcients. 68) S.K. Godunov: Mat. Sb. 47 (1959) 357–393

4.2 The Riemann Problem for Linear Systems

4.2 The Riemann Problem for Linear Systems

Recall problem (1.19), i.e., ∂t V + A∂x V = 0 , with a constant (m,m)-matrix A, and now prescribe the initial values ( V for x < 0 V0 (x) = Vr for x > 0

(4.3)

where the constant vectors V and Vr are different from each other. Using the origin as the position of the jump causes no loss of generality. Again assume the problem to be strictly hyperbolic, so that the eigenvalues λi of the matrix A are real and different from each other. The eigenvectors belonging to these eigenvalues are then obviously real too, and linearly independent. Thus, they can be used as a basis in Rm . In areas where the solution is smooth, (1.21) and (1.22) yield: V(x, t) =

m X

vˆi0 (x – λi t)si

(4.4)

i=1

with ˆ 0 (x) = (ˆ V v10 (x), vˆ20 (x), . . . , vˆm0 (x))T = S–1 V0 (x) . In particular, this leads to ˆ 0 (x) =: (w1 , w2 , . . . , wm )T = S–1 V V

for x < 0

and analogously to ˆ 0r (x) =: (˜ ˜2, . . . , w ˜ m )T = S–1 Vr w1 , w V

for

x>0

˜i . with certain constants wi and w If the eigenvalues are ordered as in (3.30), V(x, t) =

k0 X i=1

˜ i si + w

m X

wi si

(4.5)

i=k0 +1

follows from (4.4), provided that for a given position (x, t) the ﬁrst k0 eigenvalues are the particular ones for which x – λi t > 0 (i = 1, 2, . . . , k0 ), whereas x – λi t < 0 (i = k0 + 1, k0 + 2, . . . , m) holds for the other m – k0 eigenvalues. Let us now look at the i-characteristics x = λi t arising from the jump position, and allow t0 > 0 to be ﬁxed. Let us move for this ﬁxed t0 from the left to the right along a line parallel to the x-axis. This line crosses the k0 -characteristic at the position ˜ k0 in Eq. (4.4) x0 = λk0 t0 (cf. Fig. 4.2). As soon as this happens, the coefﬁcient w must be replaced by wk0 . The solution V described by (4.5) then obviously jumps. The height of this jump is ˜ k0 )sk0 [V ]k0 (x0 , t0 ) = (wk0 – w

(4.6)

75

76

4 The Riemann Problem

Fig. 4.2 How to solve the linear Riemann problem.

and leads to ˜ k0 )λk0 sk0 = λk0 [V ]k0 (x0 , t0 ) . A[V ]k0 (x0 , t0 ) = (wk0 – w

(4.7)

We already know that discontinuities move along characteristics in linear problems. This makes the k0 -characteristic line through the jump position a k0 -shock ˜ k0 , meaning that (4.7) simply becomes the Rankine–Hugoniot condiΓk0 if wk0 =/ w tion (2.15) applied to the special linear situation. Equations (4.5) and (4.7) yield the result that the solution to our Riemann problem for t > 0 can also be formulated as X ˜ i )si V(x, t) = VL – (wi – w (4.8) i

λi < xt

or as V(x, t) = VR +

X i

λi >

˜ i )si (wi – w

x t

(cf. Fig. 4.3).

Fig. 4.3 Characteristic cones of the solution to the linear Riemann problem.

(4.9)

4.3 The Aw–Rascle Trafﬁc Flow Model

4.3 The Aw–Rascle Trafﬁc Flow Model

The Riemann problem plays a signiﬁcant role in a continuous trafﬁc ﬂow model which is more realistic than the Lighthill–Whitham model used in Chapter 2 to demonstrate the loss of uniqueness that can occur for weak solutions. This more realistic mathematical description of trafﬁc ﬂow was created by A. Aw and M. Rascle. 69) The Lighthill–Whitham model does not respect the fact that a car – in contrast to gas particles – is only stimulated by the trafﬁc ﬂow situation in front of it, whereas a gas particle is also inﬂuenced by the situation behind it. Taking this important difference into account, but retaining the idea of a continuous model, Aw and Rascle developed a model that leads to a system of two equations, namely to ∂t ρ + ∂x (ρv) = 0 ` ´ ` ´ ∂t v + p(ρ) + v∂x v + p(ρ) = 0 ,

(4.10)

where p(ρ) is an increasing function with p(ρ) ~ ργ near ρ = 0 (γ > 0), as well as for all ρ with ρp

(ρ) + 2p (ρ) > 0 .

(4.11)

This structure of the “pressure” p results from the fact that knowledge of this function is only of interest close to the vacuum, and the fact that ρp(ρ) is strictly convex. Equation (4.10) is strictly hyperbolic for ρ > 0 because it can be written as ∂t U + A(U)∂x U = 0

(4.12)

with U = (ρ, v)T and A(U) =

v 0

! ρ v – ρp (ρ)

with two real eigenvalues which are obviously different from each other as long as ρ > 0. The principles developed by Aw and Rascle after observing real trafﬁc ﬂows are as follows: 1. The system is expected to be strictly hyperbolic 2. The Riemann problem with arbitrary bounded nonnegative Riemann data U must lead to a solution where ρ and v are nonnegative 3. If U± are the data for this Riemann problem, all waves connecting any state U to its left must lead to a shock speed λ u v 69) A. Aw, M. Rascle: SIAM J. Appl. Math. 60 (2000) 916–983

77

78

4 The Riemann Problem

4. Braking leads to shock waves, while acceleration leads to rarefaction waves that fulﬁll principle 3 5. There must be no continuous dependence of the solution to the Riemann problem on the initial data at ρ = 0.

79

5 Real Fluids 5.1 The Navier–Stokes Equations Model

An ideal ﬂuid was deﬁned as being a material without friction between neighboring particles with different ﬂow velocities, and thus without tangential forces along the surfaces of arbitrary volumes W(t) picked out of the ﬂuid ﬂow. The mathematical model of ideal ﬂuids can therefore only describe certain situations, e.g., low-speed ﬂows, those with low densities, low viscosities or high temperatures, etc., and the results obtained from it can seriously contradict reality if these properties are not ensured. The force acting on a bridge pier parallel to the ﬂow of a calmly ﬂowing stream was mentioned when the Kutta–Zhukovsky formulas were presented in Section 1.3. Thus, if the tangential forces generated by the viscosity play an important role in a real ﬂow situation, they cannot be neglected in a mathematical model that is expected to describe the ﬂow in a satisfactory manner. Let us therefore try to revise the Euler equations model of ideal ﬂuids by taking the viscosity forces into account too so that we can generate a model of real ﬂuids. Recall that in our scenario, viscosity always refers to friction between ﬂuid particles, 70) and so these additional forces can only occur in the case of motion (i.e., for a nonvanishing ﬂow velocity), in which case they lead to an additional term within the law of the conservation of momentum. The right hand side of (1.5) consisted of the forces per unit volume k only, and so the forces for the volume W(t) are therefore given by Z

k d(x, y, z) . W(t)

The frictional forces generated by the ﬂuid particles that are outside this volume but moving along its surface ∂W(t) are obviously proportional to the area of ∂W(t) and are directed tangentially, whereas the forces generated by the pressure of the external ﬂuid are proportional to this area, too, but are directed perpendicular to ∂W(t). 70) See (1.54) and its explanation

80

5 Real Fluids

If s = (s1 , s2 , s3 )T are the forces per unit area that are generated by the outer ﬂuid and act on the surface ∂W(t) of an inner volume W(t) arbitrarily picked out of the ﬂuid, the relation s = s(t, x, y, z, n) ,

(x, y, z) ∈ ∂W(t)

(5.1)

holds, where n is the outwardly directed unit vector normal on ∂W(t) at (x, y, z) ∈ ∂W(t). Here, s depends linearly on n: s = (σij )n

(5.2)

with σij : (x, y, z, t) → σij (x, y, z, t). Equation (5.2) follows from the fact that, within the law of the conservation of momentum, now written as ﬁ ﬂ „ « ﬀ Z j 1 1 ∂t q + q, ∇ q + div q q d(x, y, z) ρ ρ W(t) Z Z k d(x, y, z) + s do , = W(t)

∂W(t)

the volume terms vanish to a higher order than the surface terms when W(t) → 0. Here, the stress tensor (σij ) is treated like a real (3, 3)-matrix. Moreover, Euler’s law of vanishing momentum, i.e., Z Z [r, k] d(x, y, z) + [r, s] do = 0 , (5.3) W(t)

∂W(t)

r = (x, y, z)T , yields the symmetry of the stress tensor: σij = σji .

(5.4)

In the case of ideal ﬂuids, vanishing friction must be modeled by σij = 0 for i =/ j. Taking (5.2) into account, this leads to σ11 = σ22 = σ33 =: –p

(pressure)

because the vectors s and n are parallel in this situation. This is a slightly more general version of (1.5). When real ﬂuids are of interest, the terms σij (i =/ j) can be expected to be small, and the terms σii will certainly differ from –p by only a small amount. We therefore put ( σii + p for i = j ˜ij = (5.5) σ for i =/ j , σij ˜ij are small. so that all of the σ

5.1 The Navier–Stokes Equations Model

Because of the symmetry of (σij ), the tensor (˜ σij ) is also symmetric. If the exterior forces k per unit volume are replaced by the forces ˆk per unit mass, ﬁ ﬂ „ « ﬀ Z j 1 1 ∂t q + q, ∇ q + div q q – ρkˆ d(x, y, z) ρ ρ W(t) Z Z (˜ σik ) n do pn do + =– ∂W(t)

∂W(t)

follows. Obviously, if a ﬂuid does not move or if neighboring particles do not move with different velocities, friction cannot occur. In modeling the ﬂow, we therefore as˜ij (i, j = 1, 2, 3) depend only on the components of the vectors sume that the terms σ ∇ui (i = 1, 2, 3). Here, we restrict ourselves to so-called Newtonian ﬂuids, i.e., the ˜ij depend on the components ∇ui homogeneously and linearly: terms σ 3 X

˜ij = σ

(i, j)

αμν ∂xμ uν .

(5.6)

μ, ν=1

˜ij is a function of the elements of the Jacobian Ju: Thus, every σ ˜ij (Ju) , ˜ij : Ju → σ σ (i, j)

and the αμν are constants. Analogously to the case of ideal ﬂuids, where every axis is a main stress axis, we also expect relation (5.6) to be independent of orthogonal linear transformations in our model, i.e., ` ´ ` ` T ´´ ˜ij A = σ ˜ij A JuA , AT σ ∀ A ∈ R(3,3) with AT A = I . (5.7) It is trivial to show that the matrix B := div u · I is symmetric; it is also a diagonal matrix whose diagonal elements equal each other, so it shows the property AT BA = B . The elements bij of B depend on the elements of the Jacobian, i.e., B = (bij (Ju)). Because div u = trace Ju and because traces of matrices are invariant to orthogonal transformations, ` ` T ´´ ` ´ bij A JuA = bij (Ju) = B also follows. Thus, B fulﬁlls all of the requirements for (˜ σij ). However, D := Ju + (Ju)T =: D (Ju) is also symmetric, it depends on the elements of the Jacobian in a homogeneous and linear way, and it shows the property

i.e.,

AT DA = AT JuA + AT (Ju)T A ` ´T = AT JuA + AT JuA , ` ´ AT D(Ju)A = D (AT JuA) .

Thus, D also fulﬁlls all of the demands on (˜ σij ), as does λB + ηD with the arbitrary constants λ, η. It can be shown that all of these types of matrices satisfy our

81

82

5 Real Fluids

demands. Hence, ﬁ ﬂ „ « ﬀ Z j 1 1 ∂t q + q, ∇ q + div q q – ρkˆ d(x, y, z) ρ ρ W(t) Z (–pI + λB + ηD) n do = 0 . – ∂W(t)

By means of the divergence theorem, Z

Z

pn do

∇ p d(x, y, z)

=

∂W(t)

W(t)

Z

Z

Bn do

and ∂W(t)

= W(t)

0

1 div b1 B C @div b2 A d(x, y, z) div b3

follow, where the bi are the row vectors of B, i.e., Z Z Bn do = ∇(div u) d(x, y, z) , ∂W(t)

W(t)

and similarly Z

Z

Dn do = ∂W(t)

W(t)

0 1 div d1 B C @div d2 A d(x, y, z) div d3

where div d1 = 2∂x (∂x u1 ) + ∂y (∂y u1 + ∂x u2 ) + ∂z (∂z u1 + ∂x u3 ) = Δu1 + ∂xx u1 + ∂xy u2 + ∂xz u3 = Δu1 + ∂x (div u) . Analogous formulae hold for div d2 and div d3 . Equation (5.8) therefore becomes Z j

ﬁ

∂t q + W(t)

ﬂ „ « 1 1 q, ∇ q + div q q – ρkˆ ρ ρ ﬀ

+ ∇p – λ∇(div u) – ηΔu – η∇(div u)

d(x, y, z) = 0

for each part W(t) picked arbitrarily out of the volume of the ﬂuid, i.e., ∂t (ρu) + u, ∇(ρu) + (ρ div u)u – ρkˆ + ∇p – (λ + η)∇(div u) – ηΔu = 0

(5.8)

5.1 The Navier–Stokes Equations Model

or ∂t ρu + ρ∂t u + u, ∇ρu + ρu, ∇u + (ρ div u)u – ρkˆ + ∇p – (λ + η)∇(div u) – ηΔu = 0 . Because of ρ div u + u, ∇ρ = div(ρu), the relation {∂t ρ + div(ρu)} u + ρ∂t u + ρu, ∇u – ρkˆ + ∇ρ – (λ + η)∇(div u) – η Δu = 0 follows. Taking the continuity equation into account, and dividing by ρ, we then obtain the so-called Navier 71)–Stokes 72) equations: ∂t u + u, ∇u –

λ+η η 1 ∇(div u) – Δu = – ∇ p + kˆ ρ ρ ρ

(5.9)

Here, η is called the ﬁrst viscosity coefﬁcient. Of course, historically speaking, there were several arguments that led to the Navier–Stokes equations, which is why λ is not called the second viscosity coefﬁcient, but rather ξ := λ + 23 η. These coefﬁcients are constants that depend on the ﬂuid used as well as its temperature. The unit of measurement is g · cm–1 · s–1 = Poise (cf. footnote 74). When considering incompressible ﬂows characterized by div u = 0, λ does not occur in the equations, and if the exterior forces kˆ do not play a role either, the Navier–Stokes equations are reduced to ∂t u + u, ∇u – νΔu +

1 ∇p = 0 , ρ

(5.10)

η is called the kinematic viscosity. ρ By similar arguments, the law describing the conservation of energy takes the following form for viscous ﬂows: ““ ” ” ∂t E + div ((E + p) u) – λ · div(div u · u) – η · div Ju + (Ju)T u = 0 .

where ν =

Some of the terms within this equation can be simpliﬁed, e.g., div(div u · u) = ∇(div u), u + (div u)2 etc. The Navier–Stokes equations model has been shown by experiments to yield excellent results. A feel for the magnitude of kinematic viscosity and its sensitivity to temperature is given by Table 5.1. 71) Claude Louis Marie Henri Navier (1785–1836); Dijon, Paris 72) George Gabriel Stokes (1819–1903); Cambridge

83

84

5 Real Fluids Table 5.1 Values of the kinematic viscosity. Fluid

Temperature [◦ C]

Kinematicˆ ˜ viscosity cm2 /s

Mercury

0 10 20 0 10 20 0 10 20 0 20

0.00125 0.00123 0.00117 0.133 0.140 0.143 0.0178 0.0130 0.0101 7.34 3.82

Air Water Machine oil (brand-dependent; approx. values shown)

In order to experimentally study the behavior of a real ship in the ocean or a wing passing through the air, etc., engineers normally use a small model of the ship that is dipped into a wave canal or a small model of the wing that is placed in a wind tunnel, respectively. Let us attempt to ﬁnd the conditions under which the results of the experiments agree with reality. To answer this question, it is advisable to use dimensionless quantities. For convenience, let us consider the case of an incompressible ﬂow. Assume that u0 = (u0 , 0, 0)T is the ﬂow velocity measured far away from the solid, i.e., from the hull of the ship, from the wing etc. Let L be a characteristic length of the solid, e.g., the length of the ship, the length of the wing’s airfoil, etc., let T be the time the ﬂuid needs to transit the length L at velocity u0 , and let P = ρ2 u20 be the dynamic pressure in a great distance of the solid. Let us consider the particular case where the exterior forces per unit volume can be neglected. We now introduce the dimensionless quantities ˜ u=

1 u, u0

˜t =

t , T

˜ p=

p , 2P

˜ x=

x , L

˜y =

y , L

˜ z=

z . L

The incompressible Navier–Stokes equations then become ˜u = 0 , ˜ u + ∇˜ ˜ p – 1 Δ˜ ˜ + ˜ ∂˜tu u, ∇˜ Re

(5.11)

where Re :=

ρu0 L u0 L = η ν

(5.12)

is the so-called Reynolds number and ` ´ ˜ = ∂˜x˜x + ∂˜y˜y + ∂˜z˜z . ˜ := ∂˜x , ∂˜y, ∂˜z T , Δ ∇ Thus, the experiment coincides with the real-world behavior of the solid when placed in the ﬂow if the Reynolds numbers of both situations coincide. This leaves

5.2 Drag Force and the Hagen–Poiseuille Law

a great deal of freedom in the particular choices of u0 , L, ν that can be used in the experiment. The ﬂows are then said to be similar. In the case of η =/ 0, the Laplace operator occurs in (5.10), so this system of equations now becomes a parabolic system if the ﬂow is instationary or an elliptic system if a higher dimensional stationary ﬂow is present, respectively. The continuity equation as well as the energy conservation law that is already known from the model of ideal ﬂuid ﬂows remain unaffected by the viscosity and are therefore still valid. This also holds for the additional equation of state when gas ﬂows are being investigated. There are only a few situations where the Euler equations or the Navier–Stokes equations can be explicitly solved by well-known functions. In all other situations, they must be solved approximately using numerical procedures. Two of the problems which can be treated explicitly will be studied in the next section, and both problems are examples of stationary ﬂows.

s

Remark The Navier–Stokes equations do not consider the additional buoyancy phenomenona that arise in compressible gas ﬂows from density variations caused by temperature differences between different parts of the gas. In order to account for these phenomenona, the Navier–Stokes equations have to be enriched with additional terms. If the enriched equations are also written in a dimensionless form, further dimensionless parameters besides the Reynolds number occur, in particular the Prandtl number Pr :=

ν ρcp κ

where κ is the speciﬁc thermal conductivity. An example of such a scenario is the ﬂow of hot gases through pipes, e.g., tunnel ﬁres, as studied in Section 5.7.

5.2 Drag Force and the Hagen–Poiseuille Law

Viscous ﬂows can be laminar or turbulent. The ﬂow is laminar if neighboring ﬂuid particles move more or less parallel to each other along stable trajectories, whereas turbulent ﬂows are characterized by a disordered ﬂow superimposed on the main ﬂow. Thus, the turbulent ﬂow is an instability phenomenon. When Reynolds studied the ﬂow of liquids through small glass tubes, he noticed that the transition from the laminar to the turbulent state of the ﬂow takes place in a very abrupt way, and depends on the parameter that is now called the Reynolds number (cf. (5.11)). The transition to instability will be brieﬂy discussed in the ﬁfth section of this chapter. In this section, the ﬂows are assumed to be laminar. Our ﬁrst simple application of the Navier–Stokes equations concerns a laminar viscous stationary ﬂow in the y-direction with a small velocity parallel to a given area δF of a plain solid plate, where this area is part of the (y, z)-plane. Let the ﬂuid

85

86

5 Real Fluids

be incompressible in the sense of ρ = const. Because of the friction between the moving ﬂuid particles and the ﬂuid particles that are stuck to the rigid body, in accordance with the no-slip condition, there is a force acting on δF. Obviously, this force has only one component in the y-direction, and the velocity is independent of the space variables y and z: u = (0, u(x), 0)T

with

u(0) = 0 (no-slip condition) .

The undisturbed velocity of the ﬂuid that passes over the surface area at a great distance is assumed to be constant, i.e., ∂x u(x) → 0 for x → ∞. Compared with (5.10), the forces per mass unit occuring within the Euler equations are now enriched by the additional term νΔu, so that the forces per volume unit are now modeled by 0 1 0 B C w = @η∂xx u(x)A . (5.13) 0 The resistance of the relevant area (which acts in the direction opposite to the ﬂow and arises from friction) or vice versa the drag force that acts on the plate due to the friction of the ﬂuid is therefore 0 1 Z∞ 0 B C δF w dx = @–ηux (0)δFA . 0 0 Hence, the drag related to the friction of a laminar ﬂow is given by W = η∂x u(0)δF .

(5.14)

Coming back to Reynolds’ experiments, as an example let us now study the onedimensional laminar stationary cylindrically symmetric ﬂow of an incompressible liquid through a cylindrical pipe of radius R where gravitational force is neglected, where other outer forces do not occur, and where the no-slip condition u = 0 holds at the interior surface of the pipe (cf. Fig. 5.2).

Fig. 5.1 Flow along a planar membrane.

5.2 Drag Force and the Hagen–Poiseuille Law

Fig. 5.2 Laminar pipe ﬂow.

Because of u = (u, 0, 0)T , the relations u, ∇ u = (u∂x u, 0, 0)T , Δu = (Δu, 0, 0)T

div u = ∂x u

and

follow. Taking the continuity equation div u = 0 into account, we ﬁnd that ∂x u = 0, and so ∂xx u = 0 too. Hence, (5.8) reduces to 0 1 0 1 0 1 Δu ∂ p 0 ηB C 1B x C B C (5.15) – @ 0 A + @ ∂y pA = @0A . ρ ρ 0 0 ∂z p This leads to ∂y p = ∂z p = 0 ,

i.e.,

p = p(x) ,

and therefore to Δu = ∂yy u + ∂zz u =

∂x p . η

(5.16)

The left side of (5.16) is represented by a function that depends only on the variables y and z, whereas the right side depends only on x. Therefore, ∂x p must be constant. The particular type of ﬂow considered here can therefore only occur if p depends linearly on x. If polar coordinates (r, ϕ) are introduced in the (y, z)-plane, and if the cylindrical symmetry is respected, i.e., u = u(r) independent of ϕ, the well-known transformation rule Δu =

1 1 ∂r (r∂r u) + 2 ∂ϕϕ u r r

transforms (5.16) into ∂r (r∂r u) =

∂x p r. η

87

88

5 Real Fluids

Because p is independent of (y, z), and hence independent of (r, ϕ), integration leads to r∂r u =

∂x p r2 + c1 η 2

(5.17)

with an integration constant c1 . Besides the no-slip condition u(R) = 0, the boundary condition |(∂r u)r=0 | < ∞ must also be fulﬁlled, so c1 has to vanish. If (5.17) is now integrated, u(r) =

∂x p r2 + c2 2η 2

results, where the constant c2 must be chosen in such a way that the no-slip condition is fulﬁlled. This ﬁnally yields the so-called Hagen–Poiseuille law u(r) = –

∂x p ` 2 2 ´ R –r . 4η

(5.18)

This law exhibits a parabolic velocity distribution for every ﬁxed x, and the ﬂow follows the direction of decreasing pressure (cf. Fig. 5.3).

Fig. 5.3 The Hagen–Poiseuille ﬂow through a pipe.

We can now immediately determine the mass that ﬂows through the crosssection Q of the tube per second: Z2π ZR

Z

ρu dQ = –

M= Q

0

0

ρ∂x p ` 2 2 ´ πρ∂x p 4 R – r r dr dϕ = – R , 4η 8η

(5.19)

which is proportional to the fourth power of the radius R, though the cross-section is only proportional to the second power of the radius.

5.2 Drag Force and the Hagen–Poiseuille Law

s

Remark This mass ﬂow was experimentally discovered by the Prussian civil service hydraulic engineer Hagen 73) in 1839 without any knowledge of (5.18), and also (independently) by the French physiologist Poiseuille 74) in 1841, when he was interested in blood ﬂow. It was also Poiseuille who discovered Eq. (5.19). The parabolic velocity distribution was ﬁrst mentioned in 1845 by Stokes. The experiments of Hagen and of Poiseuille were performed with small tube diameters (0.015–3.00 mm). They conﬁrmed Eq. (5.19), and so the law (5.18) is now named after them. Reynolds used glass tubes with greater diameters (up to 10 mm); he also added some dyes to the liquids he worked with and noted that there is a critical value of the parameter which is now called the Reynolds number, and this critical Reynolds number is characteristic of the transition from laminar ﬂow to turbulent ﬂow. He found that Recr = 1160, and recognized that the Hagen– Poiseuille law loses its validity in regions of turbulence: the mean ﬂow velocity decreases due to a higher ﬂow resistance. The drag force of the laminar Hagen–Poiseuille ﬂow in a section of a cylindrical pipe of length L can be computed via (5.14) and of (5.18) in the following way: W = η2πRL (∂r u)r=R = η2πRL

∂x pR = πR2 Δp , 2η

(5.20)

where Δp denotes the pressure difference between the two ends of the particular part of the pipe under consideration. There are some other cross-sections of ducts for which the Navier–Stokes equations can be solved analytically under the same ﬂow assumptions as used in the Hagen–Poiseuille case (steady, laminar, incompressible, viscous), such as annuli, equilateral triangles, etc. 75) Situations similar to the ﬂow of a liquid through narrow ducts occur for other ﬂows through narrow gaps or slits, e.g., in investigations of the ﬂow of lubricating oils in lubrication research – termed tribology. As already mentioned, there are only a few examples where the Navier–Stokes equations can be solved exactly. This led to efforts to simplify these equations wherever possible. Some of the most important developments concerning such simpliﬁcations for small and large Reynolds numbers will be brieﬂy discussed in the next sections. Also, for large Reynolds numbers, we do not go back to the Euler equations but instead preserve some of the structures of the Navier–Stokes equations and study other suitable simpliﬁcations, and the no-slip condition will also be retained for large Reynolds numbers. The small Reynolds number case was particularly thoroughly investigated by Stokes and led to a model equation that is now called the Stokes approximation (cf. (5.25)). On the other hand, the theory and practical applications of ﬂuid dynamics for large Reynolds numbers was signiﬁcantly advanced by Prandtl 76) in his boundary layer theory and his wing theory. 73) Gotthilf Heinrich Ludwig Hagen (1797–1884); Berlin 74) Jean Louis Marie Poiseuille (1799–1869); Paris

75) C.Y. Wang: Ann. Rev. Fluid Mech. 23 (1991) 159–177 76) Ludwig Prandtl (1875–1953); Hannover, Göttingen

89

90

5 Real Fluids

5.3 Stokes Approximation and Artiﬁcial Time

The difﬁculties involved in solving stationary or instationary Navier–Stokes problems or Euler problems are mainly due to the nonlinearity of the convection term u, ∇u. Another difﬁculty, which is particularly associated with the case of an incompressible instationary ﬂow, is the fact that the pressure p that must be determined along with the velocity u is not represented in the continuity equation by its time derivative, e.g., by an additional term κ∂t p (κ = const). If it would have been represented by its time derivative, and if a suitable initial pressure p0 would have been prescribed, the instationary problem could be reformulated as a genuine initial boundary value problem with the differential equation ∂t ϕ – L[ϕ] = 0 with ϕ = (u1 , u2 , u3 , p)T

(5.21)

which could be treated theoretically as well as numerically using appropriate techniques. Let us assume that the solutions of instationary Navier–Stokes problems converge for t → ∞ in each case towards the solution of the corresponding stationary problem, independent of the initial function u0 . If this also holds for the instationary problem (5.21) consisting of the Navier– Stokes equations and of a continuity equation enriched with a physically unjustiﬁed additional time derivative of p, the solution of this manipulated problem must coincide with the stationary solution of the original Navier–Stokes problem for t → ∞ because of pt → 0 for

t→∞.

This leads to the following idea for a method of solving the stationary Navier– Stokes problem. Introduce an artiﬁcial time τ and enrich the stationary Navier– Stokes equations as well as the continuity equation by adding linear combinations of certain terms in such a way that, after the addition of a more or less arbitrary initial function, a complete instationary initial or initial boundary value problem is created with a solution that is expected to be able to nullify the additional artiﬁcial terms for τ → ∞. In engineering, this concept is called the method of artiﬁcial compressibility and is occasionally realized numerically. Such an artiﬁcially enriched stationary problem might appear as follows: 9 1 –∂τ u = u, ∇u + ∇ p – Δu + α∇ div u= in Ω , (5.22) Re ; –∂τ p = β2 div u on ∂Ω ,

u = u0 [0]

u=u

,

p = p0

for τ = 0 ,

where u[0] is assumed to coincide with u0 on ∂Ω. α > 0 and β =/ 0 are arbitrary constants. They should be chosen in such a way that the numerical treatment of the artiﬁcially enriched problem produces a method that is as effective as possible.

5.3 Stokes Approximation and Artiﬁcial Time

Of course, the expected property of the artiﬁcially added terms – that they will vanish as τ increases – should be guaranteed by a mathematical proof. From a numerical point of view, this strategy can lead to an iterative procedure for approximately solving the stationary boundary value problem in the following way. Use the more or less arbitrarily added artiﬁcial initial function as an initial approximation, i.e., ϕ

[0]

:=

u[0] p0

!

.

(5.23)

Then discretize (5.22) with respect to the artiﬁcial time variable τ, using a suitable step size Δn τ in each step, e.g., an equidistant step size Δτ. An example of an actual semidiscrete approximate equation is the explicit Euler method ˆ ˜ ϕ[n+1] = ϕ[n] – Δn τ · L ϕ[n]

(n = 0, 1, 2, · · · )

(5.24)

formulated in the style of (5.21). Here, every u[n] is expected to fulﬁll the boundary conditions given in (5.22). ϕ[n] will then not only be interpreted as an approximate solution of the artiﬁcial instationary problem at its nth artiﬁcial time step, but also as an approximate solution of the stationary problem after the nth iteration step. If the iteration converges sufﬁciently fast, it will be stopped after a certain number of steps n0 , and ϕ[n0 ] (x, y, z) will then be considered the ﬁnal approximate solution to the stationary problem. Initially, this method was only used and justiﬁed heuristically by engineers, who asked the mathematicians for a mathematical justiﬁcation too. This justiﬁcation could then – at least partially – be provided by a proof of the fact that, for the Stokes approximation, the solution of (5.22) actually converges for τ → ∞ to the solution of the stationary problem. 77) Here, the truncation errors introduced by additional space discretizations are ignored. The Stokes approximation is developed from the Navier–Stokes equations by omitting the convective term. This seems a reasonable omission for incompressible ﬂows with small Reynolds numbers. Namely, if (5.11) is written as ˜u ˜ u + Re ·∇ ˜ ˜=0, Re ·∂˜t˜ u + Re ·˜ u, ∇˜ p–Δ one can see that the linear diffusion term dominates over the convection term. If we omit the tildes and restrict ourselves to the stationary problem, only Re ·∇ p – Δu = 0

(5.25)

remains upon ignoring the convection term. Finally, we replace Re ·p with a new quantity p and enrich the system in the way presented in (5.22) for example. The 77) St. Fellehner: Dissertation, Hamburg 1995

91

92

5 Real Fluids

system is then found to take the form ) ∂τ u + ∇ p = Δu + α∇ div u ∂τ p + β2 div u = 0

in Ω ,

(5.26)

u = 0 on ∂Ω , u = u[0] ,

p = p0

for τ = 0 ,

where the linear structure of (5.25) is also taken into account. Because of this structure, it is sufﬁcient to show the convergence of u for increasing values of τ to the stationary solution in the case of vanishing exterior forces and homogeneous boundary conditions. The velocity part of the stationary solution of (5.26) then obviously vanishes, i.e., u=0, and the following theorem holds: Theorem 5.1 The solution u of (5.26) 78) converges in the sense of the L2 -norm for τ → ∞ to the zero function. 79) Proof: Using the notation Ju · v = (∇ u1 , v, ∇ u2 , v, · · · , ∇ ud , v)T , X Ju, Jv = ∂xj ui ∂xj vi , Z

i,j

(p, q) =

√

pq dx ,

pΩ =

u, v dx ,

uΩ =

Ju, Jv dx ,

JuΩ =

(p, p)

in L2 (Ω) ,

Ω

Z

(u, v) =

√

(u, u)

in Ld2 (Ω) ,

Ω

Z

(Ju, Jv) =

√

(Ju, Ju)

in Ld~d 2 (Ω) ,

Ω

taking div(Ju · v) = Δu, v + Ju, Jv , into account or – using Gauss’ divergence theorem – the relation ﬂ Z ﬁ ∂u , v do – (Ju, Jv) , (Δu, v) = ∂n

(5.27)

∂Ω

and ﬁnally also taking into account the fact that the integral along the bound78) For convenience, we omit considerations concerning the behavior of p. 79) Here, it is assumed that there is a unique smooth solution to the artiﬁcial instationary problem.

5.3 Stokes Approximation and Artiﬁcial Time

ary vanishes, (5.26) leads to „ « 1 d 1 1 u2Ω + 2 p2Ω = (u, uτ ) + 2 (p, ∂τ p) 2 dτ β β = (u, –∇ p) + (u, Δu) + (u, α∇(div u)) ´ 1 ` + 2 p, –β2 div u β

(5.28)

= –Ju2Ω – αdiv u2Ω . Thus, σ(τ) :=

„ « 1 u2Ω + 2 p2Ω β

decreases monotonously, and more strongly as α increases. 80) Hence, uΩ and pΩ are bounded such that σ∞ := lim σ(τ) τ→∞

exists. The Poincaré inequality 81) then yields in Ld2 (Ω) u2Ω u c1 Ju2Ω where the constant c1 > 0 depends only on the domain Ω. Equation (5.27) therefore leads to „ « d 1 σ (τ) = u2Ω + 2 p2Ω u –cu2Ω (5.29) dτ β 2 > 0. c1 Formula (5.29) implies

where c =

Z∞

σ(τ0 ) – σ∞ v c

u2Ω dτ

(5.30)

τ0

so that, in particular, the integral on the right hand side exists, i.e., uΩ ∈ L2 (R+ ). However, ´ d ` u2Ω dτ

is also bounded, as can be seen by the following arguments. 80) This can also be used for numerical purposes. 81) See, for example, R. Dautrey, J.-L. Lions: Mathematical analysis and numerical methods for science and technology, vol. 2. Berlin: Springer 1988, p. 126

93

94

5 Real Fluids

Differentiate the differential equation in (5.26) with respect to τ. This leads in a ﬁrst step that is analogous to (5.29) to d dτ

„ « 1 ∂τ u2Ω + 2 ∂τ p2Ω u –c∂τ u2Ω , β

(5.31)

and hence to the boundedness of ∂τ u. Because of ˛ ˛ ˛ d ` ´˛ 2 ˛ ˛ u Ω ˛ = 2|(u, ∂τ u)| u 2uΩ ∂τ uΩ , ˛ dτ

the existence of a constant ω with ˛ ˛ ˛ d ` ´˛ 2 ˛ ˛ u Ω ˛ dτ ˛uω

(5.32)

follows. The inequalities (5.32) and (5.30) then imply lim u(τ)Ω = 0 .

τ→∞

Otherwise there would be an ε > 0 and a number τi > ˜τ for every ˜τ > 0 with u(τi )2Ω v ε, and by means of the inequality –ω(τ – τi ) u u(τ)2Ω – u(τi )2Ω u ω(τ – τi ) , which follows for every τ > τi from (5.32), u(τ)2Ω v ε – ω(τ – τi ) would follow. Thus, τi + ωε

Z

u(τ)2Ω dτ v

Ji := τi

ε2 ω ε2 ε2 = – 2 ω 2ω 2ω

would also result. Now, after choosing a suitable subsequence of the sequence {Ji }, the intervals τi u τ u τi + ωε (i = 1, 2, · · · ) could be assumed to be disjunct, leading to X i

Z∞

Ji u

u(τ)2Ω dτ < ∞ . τ1

This ﬁnally would contradict the unboundedness of the partial sums of

P i

Ji .

5.4 Foundations of the Boundary Layer Theory and Flow Separation

5.4 Foundations of the Boundary Layer Theory and Flow Separation

Let us now look at the other extreme situation for the Navier–Stokes equations written in the dimensionless form (5.11), namely the case of large Reynolds numbers, which was studied by Prandtl in particular. Here, we again omit the tildes used in (5.11), and because the Reynolds numbers are large, we can expect the solutions to behave in relatively similar way to the solutions of the Euler equations. 82) Prandtl published his results in 1904 at the International Congress of Mathematicians in Heidelberg. 83) Let us now discuss his ideas. Assume that a solid is placed in a viscous ﬂow. Let the no-slip condition (1.54) hold along the wall Γ of this solid, i.e., uΓ = 0, omit any inﬂuences of exterior forces, and assume that the ﬂow was irrotational before the solid was placed in it. Thus, at this moment the velocity u could be derived from a potential Φ which fulﬁlls the potential equation ΔΦ = 0 because of the incompressibility of the ﬂuid. This leads to the fact that there is no difference from the frictionless situation. Prandtl restricted his considerations to a piece Γ of the surface of the solid which can be assumed to be (more or less) planar, 84) and this plane was then used as the (x, y)-plane in an (x, y, z)-coordinate system. At a great distance from the plane, theoretically for z → ∞, Prandtl expected the ﬂow to retain its character as a potential ﬂow, so that the Euler equations hold for large values of z. The idea that the ﬂow could perhaps be irrotational everywhere outside the solid immediately leads to a contradiction for u = 0, because the problem ΔΦ = 0

outside the solid

uΓ = 0

no-slip condition

only has the solution u = 0. We omit the proof and consider that a jump from the velocity u = 0 along Γ to a potential ﬂow of velocity u =/ 0 outside the solid is also impossible, because the existence of the viscosity term does not allow discontinuities of this type. Hence, the ﬂow between the surface of the solid and the distant parts of the ﬂow cannot be irrotational. The domain of this rotational ﬂow is called the boundary layer. Here, the word distant has a rather vague meaning from the point of view of real applications. An obvious question is: what is the thickness δ of the boundary layer (i.e., what is distance at which the difference between the real ﬂow and the 82) It took many years to prove mathematically that the Navier–Stokes solutions are close to the Euler solutions for small values of η: K. Nickel showed in Arch. Rat. Mech. Anal. 13 (1963) 1–14 that both solutions can, under certain conditions and for η → 0, be asymptotically considered to be solutions of the Prandtl boundary layer equations (5.48) presented in this section. This also subsequently justiﬁed Prandtl’s

assumption regarding the small diameter of the boundary layer that we will use later. 83) In: Verh. d. III. Int. Math.-Kongr. Heidelberg 1904. Leipzig: Teubner 1905, pp. 484–494 84) W. Tollmien showed that all of these considerations also hold for curved surfaces provided that the curvature does not vary too much; see Handb. d. Exper.-Physik IV, I (1938) 248ff.

95

96

5 Real Fluids

original potential ﬂow can be neglected)? Of course, the answer to this depends on a convention. Often, a 1% difference between the velocities is an accepted value. Prandtl assumed that the thickness δ was small compared with the length L used in order to write the Navier–Stokes equations in the dimensionless form (5.11), i.e., δ << 1 ,

(5.33)

and he assumed that the transition from rotational ﬂow to irrotational potential ﬂow is continuous. Let all of the derivatives occurring in the following considerations exist; let them be continuous; and let the dimensionless velocity U of the distant potential ﬂow be parallel to Γ, i.e., parallel to the (x, y)-plane. Hence, ` ´T U = U(x, y, t) = U(x, y, t), V(x, y, t), 0 . Moreover, the components of U are assumed to be of the same order of magnitude as the velocity u0 used to derive the dimensionless form (5.11) of the Navier–Stokes equations, so that U and V are of the order of magnitude 1. Because of the continuous transition from u = (u, v, w)T to U, the assumptions u = O(1) and

v = O(1)

for z → δ

(5.34)

with Landau symbol O are reasonable, and this will certainly also be true along all of the layers parallel to Γ, particularly for x, y → 1 where x and y are the dimensionless spatial variables in (5.11). Concerning the spatial derivatives of the u- and v-components parallel to Γ, as well as their time derivatives, Prandtl expected that they would not vary very much within the boundary layer, i.e., ∂t u, ∂t v, ∂x u, ∂y u, ∂xx u, ∂yy u, ∂x v, ∂y v, ∂xx v, ∂yy v = O(1) .

(5.35)

If the continuity equation div u = 0 is also taken into account, (5.35) yields a further result, namely ∂z w = O(1)

for z → δ

or for x, y → 1 ,

(5.36)

so that the no-slip condition leads to ˛ z ˛ ˛Z ˛ ˛ ˛ |w(x, y, z, t)| = ˛˛ ∂z w(x, y, ζ, t) dζ˛˛ u ∂z w∞ |z| , ˛ ˛ 0

i.e., w = O(δ) .

(5.37)

Combining assumption (5.33) with the no-slip condition, the mean value theorem leads inside the boundary layer to u(x, y, z, t) W ∂z u(x, y, z, t) · z ,

5.4 Foundations of the Boundary Layer Theory and Flow Separation

and therefore with (5.34) to „ « 1 . ∂z u = O δ

(5.38)

Analogously, the relations „ « „ « 1 1 , ∂zz u = O , ∂z v = O δ δ2

„

∂zz v = O

1 δ2

«

(5.39)

also hold. Using the mean value theorem, we also obtain the statement w(x, y, z, t) = w(0, y, z, t) + ∂x w(ϑx, y, z, t) · x,

0<ϑ<1,

and because x can grow to an order of magnitude of 1, (5.37) yields ∂x w = O(δ)

(5.40)

and analogously ∂y w, ∂xx w, ∂yy w = O(δ) ,

∂t w = O(δ) .

Moreover, taking (5.36) into account, we additionally ﬁnd „ « 1 . ∂zz w = O δ

(5.41)

(5.42)

Let us now write down the three components of (5.11) separately. We will write each of the terms occuring in the equations, and below them we will note down their particular orders of magnitude (as far as they are already known). Here we follow the representation used by H. Schlichting in the ﬁrst edition of his book on boundary-layer theory: 85) 8 1 > ∂ u + u ∂x u + v ∂y u + w ∂z u – Re (∂xx u + ∂yy u + ∂zz u) > > t > 1 1 > 1 1 1 1 1 δ 1 1 > δ δ2 > > > > > > > 1 > < ∂t v + u ∂x v + v ∂y v + w ∂z v – Re (∂xx v + ∂yy v + ∂zz v) 1 1 1 1 1 1 δ δ1 –∇p = 1 1 δ2 > > > > > > 1 > ∂t w + u ∂x w + v ∂y w + w ∂z w – Re (∂xx w + ∂yy w + ∂zz w) > > > 1 > > δ 1 δ 1 δ δ 1 δ δ > δ > : (5.43)

85) H. Schlichting: Grenzschichttheorie. Karlsruhe: Braun 1951

97

98

5 Real Fluids

Of course, the pressure gradient parallel to the plane is assumed to be bounded in all real applications, i.e., ∂x p, ∂y p = O(1) . However, this implies that the ﬁrst two equations can only be valid with respect to the order of magnitude if „ « „ « 1 1 √ , i.e. δ = O , (5.44) Re = O δ2 Re and, because of the third equation, this leads to ∂z p = O(δ) .

(5.45)

Equation (5.44) shows that Prandtl’s assumption of a small thickness for the boundary layer for large Reynolds numbers does not lead to a contradiction. It also shows that the boundary layer theory can only be accepted as an approximation for the Navier–Stokes equations in the case of large Reynolds numbers. Because of ˛ z ˛ ˛Z ˛ ˛ ˛ ˛ |p(x, y, z, t) – p(x, y, 0, t)| = ˛ ∂z p(x, y, ζ, t) dζ˛˛ u |z| ∂z p ∞ , ˛ ˛ 0

the pressure differences in the z-direction within the boundary layer are of order δ2 , and are thus very small for large Reynolds numbers. We therefore put ∂z p = 0 , so that, approximately, p does not depend on z within the boundary layer: p = p(x, y, t) .

(5.46)

The pressure gradient in the boundary layer thus coincides with the well-known pressure gradient of the potential ﬂow outside the boundary layer where the Euler equations hold. Using dimensionless variables and taking W = 0 into account, we therefore approximately obtain ∂t U + U∂x U + V∂y U = –∂x p ∂t V + U∂x V + V∂y V = –∂y p

(5.47)

so that ∂x p and ∂y p are known quantities. If all of the terms in (5.43) of order δ are now neglected compared with the terms of order 1, the system of three equations reduces to only two equations, and in dimensionless form these equations are: 1 ∂t u + u∂x u + v∂y u + w∂z u – ν∂zz u = – ∂x p ρ 1 ∂t v + u∂x v + v∂y v + w∂z v – ν∂zz v = – ∂y p ρ

(5.48)

5.4 Foundations of the Boundary Layer Theory and Flow Separation

(5.48) are the Prandtl boundary layer equations with the incompressibility condition ∂x u + ∂y v + ∂z w = 0 .

(5.49)

The velocity component w remains in the system (5.48), but is now only a parameter which will later be ﬁxed by (5.49) and by boundary conditions. Such boundary conditions may, for example, be: u = v = w = 0 for z = 0 0 1 0 1 u U(x, y, t) B C B C @ v A = @ V(x, y, t) A for z → ∞ . w 0 z → ∞ and δ << L seem to be contradictory, but one must note that the boundary layer is inﬁnitely thick from a theoretical point of view, while their effective thickness from a physical point of view is very small. If the boundary layer is affected by sucking off or blowing up parts of it, the boundary conditions must model these effects, e.g., by replacing the equation w = 0 in (5.50) by w = w0 (x, y, t)

for z = 0 ,

provided that the direction of the sucking or blowing coincides with the w-direction and that its ﬂow velocity w0 is of order O(δ), i.e., „ « 1 . (5.50) w0 = O √ Re Otherwise some of the terms that Prandtl expected to be very small can no longer be neglected. As well as boundary conditions, initial conditions must also be formulated, and not just with respect to time in the case of instationary problems, but also with respect to space too in every case, because the boundary layer equations are parabolic. This parabolicity often includes simpliﬁcations when the construction of approximate analytic solutions of the Navier–Stokes equations is desired. However, Prandtl’s theory can often make life easier numerically. If the problem to be treated concerns an instationary ﬂow, the initial task is to formulate initial conditions with respect to time, i.e., conditions of the type u(x, y, z, 0) = f1 (x, y, z) v(x, y, z, 0) = f2 (x, y, z)

(5.51)

and in each case with respect to space too, i.e., to stipulate suitable initial proﬁles for the u- and v-components of the ﬂow velocity. There is no general consensus about what the initial proﬁles should look like, but it should be noted that the inﬂuence of the initial proﬁles decreases as the distance

99

100

5 Real Fluids

from the area where these proﬁles were prescribed increases. This follows from the parabolic situation. These spatial initial conditions may take into consideration that a boundary layer comes into being only when it is a certain distance from the edges of the solid placed in the ﬂuid, e.g., from the front edge of a wing: u(x0 , y, z, t) = g11 (y, z, t) v(x0 , y, z, t) = g12 (y, z, t)

(5.52)

and u(x, y0 , z, t) = g21 (x, z, t) v(x, y0 , z, t) = g22 (x, z, t) .

(5.53)

Of course, the conditions prescribed should go well together, e.g., f1 (x0 , y, z) = g11 (y, z, 0)

(5.54)

etc. Many ﬂow phenomenona that place along the surface of a solid can be quantitatively explained via the boundary layer concept. One of them is the separation of the boundary layer ﬂow from a curved surface. Here, we are going to describe this separation brieﬂy and only qualitatively in the case of stationary 2-D ﬂows. Assume that an ideal ﬂuid is ﬂowing along a curved surface. A lateral ﬂow with a main ﬂow velocity u = (u∞ , 0, 0)T , u∞ > 0 around a circular cylinder of inﬁnite length whose axis coincides with the z-axis is a 2-D example. The velocity of the particles arriving at the surface of the cylinder increases due to Bernoulli’s equation (1.31) because they initially move against decreasing pressure. The velocity then begins to decrease because the pressure grows again. Finally, the particles move behind the cylinder with their original speed (cf. Fig. 5.4).

Fig. 5.4 Ideal ﬂow around a cylinder.

5.4 Foundations of the Boundary Layer Theory and Flow Separation

In the case of a viscid ﬂow, a certain amount of the kinetic energy of the particles will be transferred in the ﬁrst phase to friction, so there will not be enough energy available in the second phase to return the particles to their former speed. It can even happen that the particles stop and that the velocity reverses direction due to the increasing pressure; i.e., separation of the boundary layer takes place.

Fig. 5.5 Flow with separation.

In the 2-D situation, the point of separation at the curved surface, which is in the 2-D-dimensions a curve Γ, can be well described mathematically. For this purpose, let u be the particular component of the ﬂow velocity parallel to Γ and let ∂n u be the normal derivative of this component. The separation point P ∈ Γ is then obviously the point where this normal derivative vanishes (cf. Fig. 5.6): ∂n u(P) = 0 .

(5.55)

Fig. 5.6 Explanation for (5.55) in the case of 2-D boundary layer ﬂow.

Beyond the point of separation, the ﬂow loses its buoyancy effect. Hence, one tries to design a wing in such a way that the point of separation is situated as close as possible to the back edge of the wing, e.g., by sucking off the boundary layer. It should be mentioned that there is no general mathematical deﬁnition of separation in the case of 3-D ﬂows.

101

102

5 Real Fluids

Fig. 5.7 Boundary layer separation on a wing.

5.5 Stability of Laminar Flows

The beginning of the transition of a ﬂow from the laminar state to a turbulent one can be looked upon as a transition from stable behavior of the solution to the Navier–Stokes equations for this ﬂow to instability. In order to investigate this idea, for convenience, let us now consider a situation that is easy to survey, namely an incompressible 2-D ﬂow and the knowledge of a stationary average laminar ﬂow in the x-direction which has a velocity U that only depends on the second space variable y: U=

! U(y) . 0

The pressure of this ﬂow is denoted by P(x, y), and (U, P) is assumed to fulﬁll the given incompressible Navier–Stokes problem 86) and to be one part of a superposition, the other part of which consists of inﬁnitesimal and incompressible disturbances (˜ u, ˜ p) with ˜ u=

˜(x, y, t) u v˜(x, y, t)

!

and ˜ p=˜ p(x, y, t) .

This second part is not necessarily a stationary ﬂow and, besides (U, P), the complete ﬂow u=U+˜ u,

p = P+˜ p

(5.56)

is also assumed to be a solution of the given Navier–Stokes problem. 87) Because the disturbances are inﬁnitesimal, quadratic terms can be neglected compared with linear terms, so that putting (5.56) into the Navier–Stokes equa86) It is absolutely problematic to suppose such an average ﬂow to exist, but it helps to understand that turbulence is an instability phenomen.

87) This assumption is called the closing, and the treatment of the instability problem in this way is called the Reynolds averaging.

5.5 Stability of Laminar Flows

tions (5.10) and into the continuity equation then yields the relations ˜ + U∂x ˜ ∂t u u + v˜ ∂t v˜ + U∂x v˜

„ 2 « d U dU 1 1 p=ν + Δ˜ u + ∂x P + ∂x ˜ dy ρ ρ dy2 1 1 p = νΔ˜ v + ∂y P + ∂y ˜ ρ ρ

(5.57)

u + ∂y v˜ = 0 . ∂x ˜ Because the average ﬂow solves the Navier–Stokes equations on its own, (5.57) reduces to the linear equations ∂t ˜ u + U∂x ˜ u + v˜ ∂t v˜ + U∂x v˜

dU 1 p = νΔ˜ u + ∂x ˜ dy ρ 1 p = νΔ˜ v + ∂y ˜ ρ

(5.58)

u + ∂y v˜ = 0 . ∂x ˜ If the ﬁrst equation in (5.58) is differentiated with respect to y and the second one with respect to x, the pressure can be eliminated so that only one equation remains for u and v besides the continuity equation. The no-slip condition along surfaces of solids leads to additional boundary conditions. Let us now suppose that the disturbances result from the superposition of waves propagating in the x-direction, where each of them fulﬁlls (5.58). These waves can be expressed by ˜ u=

! ϕ(y) i(αx–γt) e ψ(y)

(5.59)

and are called Tollmien–Schlichting waves. Here, α ∈ C, γ ∈ C. The particular case α ∈ R describes the investigation of the temporal stability, whereas an investigation of the case α ∈ C, γ ∈ R concerns the so-called spatial stability, which is not discussed here. The continuity equation leads to ϕ(y) =

i

ψ (y) , α

and the equation that remains after eliminating ˜ p then yields (U – γ)(ψ

– α2 ψ) – U

ψ = –

iν (4) (ψ – 2α2 ψ

+ α4 ψ) . α

(5.60)

The linear homogeneous fourth-order ordinary differential equation (5.60) for ψ with complex coefﬁcients is called the Orr–Sommerfeld equation. Boundary conditions are for instance the no-slip condition ψ = 0 for y = 0 and the disappearance of the disturbances for y → ∞.

103

104

5 Real Fluids

Thus, for every ﬁxed α ∈ R, an eigenvalue problem arises where the complex phase speed of perturbation γ plays the role of the eigenvalue. The parameter ν 88) is considered to be already known from the laminar average ﬂow. The laminar ﬂow corresponds to the trivial solution ψ = 0 of problem (5.60), and one asks for nontrivial solutions that bifurcate from it. If the sign of the imaginary part of γ is negative, the disturbance decreases as time increases, and its amplitude does not depend on time if this imaginary part vanishes. However, an unstable situation occurs for sgn (Im γ) > 0 .

(5.61)

Under realistic boundary conditions for U, the Orr–Sommerfeld equation normally not be solved explicitly, though it is linear and homogeneous. Numerical procedures known from numerical bifurcation theory must be used. A very thorough treatment of the Orr–Sommerfeld equation with boundary conditions corresponding to the particular situation of a half-plane ﬂow along a wall was given by J.-R. Lahmann and M. Plum. 89) The smallest value of the Reynolds number that gives validity to (5.61) is called the critical Reynolds number Recr , and is of interest because it is the smallest Reynolds number for which the ﬂow can become unstable. The curve sgn(Im γ) = 0 obviously separates the area of instability in the (α, Re)-plane from the area of a stable laminar ﬂow and is called the curve of neutral stability. It should be noted that an unstable result from this linearized theory of disturbances does not necessarily mean that the real underlying ﬂow can already be considered to be completely turbulent. One measure accepted by engineers is the so-called e9 -rule: the ﬂow is actually a turbulent one if the small amplitude of the initial disturbance has increased by a factor e9 W 8100.

5.6 Heated Real Gas Flows

The total energy E per unit volume, consisting of stored heat plus kinetic energy, is given by ρ (5.62) E = ρcv T + u2 2 (ρ is density, cv is speciﬁc heat at constant volume, u is the velocity vector of the ﬂow). From Reynolds’ transport theorem (1.3), we ﬁnd Z Z ∂ E dW = {Et + div(Eu)} dW . (5.63) ∂t W(t)

W(t)

88) Or, if the problem is formulated in 1 dimensionless form, Re 89) J.-R. Lahmann, M. Plum: ZAMM 84 (2004) 188–204

5.6 Heated Real Gas Flows

On the other hand, the change in the total energy per unit time reads Z D Z E ∂Q ˆ u dW – ρk, pn, u do + ∂t

(5.64)

∂W(t)

W(t)

(here, Q is the heat transferred into the gas volume, kˆ is the external force per unit mass, n is the outwardly directed unit vector on the surface ∂W(t) of the volume W(t) at the instant t. 90)). In (5.64), ∂Q ∂t is represented by Z Z Z ∂Q ∇T, n do + q dW = {κΔT + q} dW (5.65) =κ ∂t ∂W(t)

W(t)

W(t)

(q is the internal production of heat per unit volume, κ is the thermal conductivity). Combination of the equations (5.62) to (5.65) leads to ” “n o ” ∂ “ ρ ρ ˆ u – div(pu) , ρcv T + u2 + div ρcv T + u2 u = κΔT + q + ρk, ∂t 2 2

i.e., to (ρcv T)t + div({ρcv T + p}u) + ρut , u + ρt

u2 2

ρ u2 u, ∇(u2 ) + div(ρu) 2 2 ˆ u . = κΔT + q + ρk, +

(5.66)

Taking the continuity equation (1.4) into account as well as the Navier–Stokes equations (5.9), the last equation (5.65) leads to (ρcv T)t + div ({ρcv T + p}u) – ρ u, ∇u, u + (λ + η) ∇(div u), u ρ +ηu, Δu – u, ∇p + u, ∇(u2 ) = κΔT + q , 2

(5.67)

i.e., to (ρT)t + div(ρTu) +

˜ 1 1 κ ˆ p div u = Δ(ρT) – TΔρ – 2∇ρ, ∇T + q – Vis , cv ρcv cv (5.68)

where the viscosity terms Vis have the form Vis =

λ+η η u, ∇(div u) + u, Δu . cv ρcv

We shall later refer to the one-dimensional situation, where (5.67) becomes (ρT)t + (ρTu)x + 90) See Section 5.1

p κ q ux = Txx + – Vis cv cv cv

(5.69)

105

106

5 Real Fluids

with Vis =

η ˆ uuxx , cv

where we introduced the quantity η ˆ , particularly η ˆ = λ + 2η in the case of compressible gases. Hence, the model of the heated gas ﬂow consists of Eq. (5.67), of the continuity equation (1.4), the Navier–Stokes equations (5.9) and of an equation of state, e.g., for ideal gases p = RρT

(5.70)

where R is the Boltzmann constant.

5.7 Tunnel Fires

As mentioned at the end of Section 5.1, a particular situation occurs if there is a ﬂow of heated gas in pipes where the heat is generated within the pipe. We say that a tunnel ﬁre occurs if the heat is produced by burning objects, resulting in a mixture of air, smoke, etc., and high temperatures. Tunnel ﬁres have unfortunately occurred fairly often in recent years in rail and road tunnels, so, as well as ﬁre brigades organizing experiments, it is in the public interest to study these events from a mathematical point of view. Several authors have tried to describe this scenario using realistic mathematical models. 91) We refer here to the continuous model developed by I. Gasser and J. Struckmeier. 92) From experience, the air or air/smoke mixture in the tunnel is homogeneous in the cross-sections of the tunnel, in the sense that there is no stratiﬁcation of the ﬂow with opposite ﬂow velocity directions in different layers of the cross-section under consideration, at least if the velocities are not very small. Hence, we can take mean values of the quantities to be considered over the cross-section at a certain instant, which enables us to treat the problem as a one-dimensional one in space, where the dimensionless space variable x(0 u x u 1) represents the longitudinal positions in the tunnel pipe. However, we must account for the fact that, in a 1D model, the reduction in pressure caused by the no-slip condition at the surface of the tunnel must be replaced by another suitable condition. We should also mention 91) See, e.g., G.B. Brandt, S.F. Jagger, C.J. Lea: Phil. Trans. R. Soc. London A 356 (1998) 2873–2906 92) I. Gasser, J. Struckmeier: Math. Meth. Appl. Sci. 25 (2002) 1231–1249

5.7 Tunnel Fires

that, in this context, we will omit the heat loss at the tunnel surface, which is usually only small. For convenience, we now introduce dimensionless, i.e., scaled parameters, variables and equations (as we did in the last part of Section 5.1). We do not distinguish between the unscaled and the scaled quantities in the text below. The reference values (e.g., ρr , ur ) by which the unscaled values will be scaled (e.g., ρscaled = ρunscaled /ρr , etc.) will be listed below. We deﬁne the Reynolds number by Re =

ρ r ur d , η ˆ

(5.71)

where d is a typical value for the cross-section of the tunnel. The 1D model for the scaled density ρ, the scaled velocity u, the scaled pressure p and the scaled temperature T then reads as follows. In dimensionless form, the continuity equation (1.4) gives ρt + (ρu)x = 0 , while the Navier–Stokes equation (5.9) leads to „ « „ « 1 1 1 d 1 = p uxx + ˆk . ut + uux + x γM2 ρ Re L ρ

(5.72)

(5.73)

In the unscaled form, the exterior forces per unit mass are found to be kˆ = –g sin α – fl

(5.74)

where α = α(x) is the given slope proﬁle of the tunnel, and g is the gravitational acceleration. In the 1D model, the (unscaled) term fl =

ξ u|u| d 2

replaces the no-slip condition (1.54), i.e., the loss of force due friction at the tunnel surface, where the quadratic velocity term takes the turbulent character of the ﬂow into consideration, and where ξ describes the surface of the tunnel and must be determined by experiment. Its order of magnitude is 10–2 to 10–1 . 93) In the scaled version, we ﬁnd: gL L u |u| kˆ = – 2 sin α – ξ . ur d 2

(5.75)

In the dimensionless 1D description, (5.68) becomes (ρT)t + (ρuT)x + (γ – 1)pux = κ¯ Txx + q¯q – M2 γ(γ – 1)¯ηuuxx 93) See, e.g., I. Gasser, R. Natalini: Quart. Appl. Math. 57 (1999) 269–282

(5.76)

107

108

5 Real Fluids Table 5.2 Reference values. Quantity

Reference value

Typical magnitude

t x Cross-sectional width u ρ p f T q R cp ηˆ κ

tr = L/ur L d

15 min 103 – 104 m 10 m

ur ρr pr fr Tr = pr /(ρr R) qr = γpr ur /((γ – 1)L)

1 m s–1 1.2 kg m–3 · m2 1 bar = 105 kg m–1 s–2 · m2 10 m s–2 300 K 105 – 106 W m–3 · m2 287 m2 s–2 K–1 1005 m2 s–2 K–1 18 ~ 10–6 kg m–1 s–1 (= 18 ~ 10–5 Poise) · m2 25 ~ 10–3 kg m s–3 K–1 · m2

with η¯ =

1 d , Re L

κ¯ =

1 η¯ , γPr

q¯ =

qr L (γ – 1) ur pr ηˆ c

r ur where γ is from (1.9), M2 = ργp , Pr = κp (cf. the last remark of Section 5.1), and r Re is from (5.71). Typical values for these parameters in our context are: 2

γ = 1.4 ,

M2 = 8.6 ~ 10–6 ,

Re = 6.7 ~ 105 ,

Pr = 0.72 .

We must also take the equation of state into account. We assume that the gas in the tunnel behaves approximately like an ideal gas, so that (5.69) holds, which leads in the scaled form to p = ρT .

(5.77)

Because the tunnel has a ﬁnite length L, boundary conditions will have to be described later, and the situation at the outbreak of the ﬁre must be respected by the initial data. Table 5.2 gives the reference values used here and their orders of magnitude. Without the bolded factor m2 , they also ﬁt the 3D case. According to the Navier–Stokes equation there is a connection between the local pressure and the ﬂow velocity at the point (x, t), and hence between the pressure and the Mach number. If we introduce the quantity ε := γM2 , which is on the order of 10–5 in the situation considered here, the expansion of p with respect to ε, i.e., p = p0 + εp1 + O(ε2 ) shows that a small Mach number asymptotic is reasonable.

(5.78)

5.7 Tunnel Fires

p0 is considered to be the mean outside pressure of the open tunnel region and is therefore assumed to be constant. It is also used as the reference value pr in Table 5.2: p0 = pr = const .

(5.79)

If one introduces (5.78) into (5.77) and neglects the small quantities of order O(ε), p0 = Tρ

(5.80)

remains. Moreover, if we put (5.78) into (5.72), (5.73) and (5.76), these equations yield, via (5.80) and by omitting the O(ε) terms, ρt + (ρu)x = 0 ,

(5.81)

1 η¯ ut + uux + p1 x = uxx + ˆk , ρ ρ „ « 1 1 + q¯q . ux = κ¯ ρ xx p0

(5.82) (5.83)

Equation (5.83) shows that, under the assumptions of this section, the divergence of the ﬂow cannot vanish. In other words, it cannot be treated as an incompressible ﬂow. From (5.76), we know that the quantities η¯ and κ¯ are very small, so that the viscous term in (5.82) and the heat conduction term in (5.83) can be omitted. If we also take (5.75) into account, we end up with the following system of equations: ρt + uρx = – ρQ ut + uux +

1 L u|u| gL p1 x = – ξ – 2 sin α ρ d 2 ur

(5.84)

ux = Q with Q=

1 γp0

„

qr L ur p0

«

(γ – 1)q .

Initial data for the velocity u(x, t) and the density ρ(x, t) must be prescribed. If one replaces the last equation of the system (5.84) by the x-derived second one (and replaces ux in the new equation by Q), and if this reformulated system together with realistic initial conditions is then associated with suitable Dirichlet boundary conditions for the pressure at the tunnel entrance and exit, by the homogeneous Neumann conditions ux (0, t) = ux (1, t) = 0 ,

∀t>0

for the velocity, and by the standard inﬂow boundary conditions ρ(0, t) = ρ0

if

u(0, t) > 0 ;

ρ(1, t) = ρ1

if u(1, t) < 0 ,

for the density, realistic results can be obtained numerically.

∀t>0

109

110

5 Real Fluids

Gasser and Struckmeier discretized system (5.84) by means of some classical explicit one-step ﬁnite difference schemes, and studied several situations. The results indicate that the model is able to perform real-time numerical experiments as well as to get qualitative insights into the ﬂow conditions during a ﬁre event. In this case, the dimensionless nature of the formulation makes it easy to look for these different situations. We refer to the results of two of Gasser’s and Struckmeier’s numerical experiments: in both cases they considered tunnels of length L = 4 km and cross-sectional area 100 m2 . The ﬁre occurs halfway between the ends of the tunnels and the effect of the heat sources amounts to 20 MW. The coefﬁcient ξ in (5.75) is expected to be of order 0.1 in both cases, and the initial velocity to be 0 ms–1 . Figure 5.8 shows the particular spatial distributions of the ﬂow velocity, the pressure, the density and the temperature 20 min after a ﬁre has broken out in a tunnel without a slope, where the pressures are 1010 mbar at the entrance and 1010.1 mbar at the exit. In Fig. 5.9, we see the situation after 10 min in a tunnel with a slope of 3%, where the pressures are 1024.4 mbar at the entrance and 1010 mbar at the exit. For the people running away from the location of the ﬁre, it is important to know the ﬂow direction of the hot gases. They will need to ﬁnd this out beforehand, so that they will head in the correct direction along the tunnel (i.e., opposite to the velocity u [m/s]

pressure p [mbar]

−0.3

1010.14

−0.4

1010.12

−0.5

1010.1

−0.6

1010.08

−0.7

1010.06

−0.8

1010.04

−0.9

1010.02

−1

0

1

2

3

4

1010

0

1

density rho [kg/m]

2

3

4

3

4

temperature T [°C]

1.4

600

1.2

500 400

1 300 0.8 200 0.6 0.4

100 0

1

2

3

4

0

0

Fig. 5.8 Tunnel without a slope 20 min after a ﬁre has broken out.

1

2

5.7 Tunnel Fires velocity u [m/s]

pressure p [mbar]

2.8

1025

2.6 1020 2.4 1015 2.2

2

0

1

2

3

4

1010

0

1

density rho [kg/m]

2

3

4

3

4

temperature T [°C]

1.3

200

1.2 150 1.1 1

100

0.9 50 0.8 0.7

0

1

2

3

4

0

0

1

2

Fig. 5.9 Tunnel with a 3% slope 10 min after a ﬁre has broken out.

direction of hot gas ﬂow) in the event of ﬁre. In other words, they need to learn how to behave in such a situation in order to avoid self-destruction through panic.

111

113

6 Proving the Existence of Entropy Solutions by Discretization Procedures Though we have already studied some properties of weak solutions of conservation laws, we are yet to prove that this type of solution exists, particularly in the case of entropy solutions. The only exceptions were some scalar problems whose solutions could be speciﬁed explicitly. We are now going to address this issue.

6.1 Some Historical Remarks

Existence proofs for solutions of ordinary or partial differential equations are often structured in the following way. In a ﬁrst step, the differential equation is discretized, as is normally done in numerical analysis in order to create numerical procedures. Thus, a ﬁnite-dimensional approximate version of the original problem that can be solved on a certain grid for each suitable step size parameter h is created. If the solution to the discrete problem is only given for a set of discrete points, it is extended to the inter-grid points by interpolation or by other techniques in such a way that it belongs to the particular function space that the solution or weak solution of the original problem is expected to belong to. This extension is called reconstruction (cf. the special case of (4.2)). Moreover, we assume that all of the computation that needs to be performed is realized theoretically without generating round-off errors. If it can be shown in a second step that a sequence L of these reconstructed approximate solutions determined for each step size parameter hj of a sequence ˘ ¯ hj of positive step-size parameters with limj→∞ hj = 0 is compact in the function space under consideration, convergent subsequences of L exist. It can often be shown in a third step that, under certain conditions, the limits of those subsequences solve the original problem. Thus, solutions exist in this case, but different convergent subsequences can lead to different solutions of the original problem, because uniqueness has not yet been ensured. A well-known example of an existence proof of this type is the particular proof of Peano’s theorem on the existence of a solution to the initial value problem y = f(x, y) ,

y(x0 ) = y0 ,

f is continuous

114

6 Proving the Existence of Entropy Solutions by Discretization Procedures

where Euler’s polygon method is used in order to construct approximate solutions in the space of functions that are continuous over a compact interval and equipped with the maximum norm. Another example is the existence proof of Courant, Friedrichs and Lewy, in which this type of proof was applied for the ﬁrst time to initial value problems for systems of quasilinear hyperbolic partial differential equations. The proof was published in their famous paper of 1928, 94) and it was in this paper that a condition concerning the step ratio was formulated in order to ensure the numerical stability of the numerical procedure, later called the CFL condition. This condition ensures that the region affecting the solution at a point Q where the solution is expected to exist is covered by the region that affects the approximate solution at Q , and this condition is extremely important when treating conservation laws by means of explicit numerical methods. 95) Of course, because Courant, Friedrichs and Lewy asked for smooth solutions, they could only provide the conditions for local existence. Under certain restrictions on the initial functions, J. Glimm provided an existence theorem for global weak solutions of conservation laws in a paper that also became famous, 96) and one of his tools was also a certain ﬁnite difference procedure. We are now going to study a general theory on discretizations of partial differential equations that can also help to prove existence theorems for conservation laws such as those that occur in ﬂuid dynamics. 97)

6.2 Reduction to Properties of Operator Sequences

In 1948, Kantorowitch published his important paper on the connections between numerical mathematics and functional analysis. 98) Particularly focusing on linear partial evolution equations, in 1956 Lax and Richtmyer provided a functional analytic framework for the construction and investigation of ﬁnite difference methods 99) (cf. Section 7.4). Their theory was later generalized in respect to various aspects. Inspired by the Lax–Richtmyer theory as well as the particular question of the discretization of differential equations, in 1972 Stummel published his general theory on discretization algorithms. 100) Though it was already rather general, it was still inﬂuenced by the linear structures underlying the given general problem. This theory was later also generalized in different ways and applied to several types of 94) R. Courant, K.-O. Friedrichs, H. Lewy: Math. Ann. 100 (1928) 32–74 95) For more details, see Section 7.3 96) J. Glimm: Comm. Pure Appl. Math. 18 (1965) 697–715 97) R. Ansorge: ZAMM 73 (1993) 239–253

98) L.W. Kantorowitch: Uspeki Mat. 3 (1948) 89–185 99) P.D. Lax, R.D. Richtmyer: Comm. Pure Appl. Math. 9 (1956) 267–293 100) F. Stummel in: J. Miller (ed.): Proceedings of the Conference on Numerical Analysis Dublin 1972. London: Academic 1973, pp. 285–310

6.2 Reduction to Properties of Operator Sequences

problems, such as ordinary or partial differential equations, integral equations, optimization problems, etc. This theory is now called the functional analysis of discretization algorithms (Vainikko). However, weak solutions of differential equations or problems whose solutions are not necessarily unique had not been included in this theory. Let us therefore, in a ﬁrst step, try to describe how discretization techniques for problems of this type are embedded into the general theory, and to look for problems given in the following general form. Let X, Y, Xn , Yn be topological, usually metric spaces. We now ask for an element v ∈ X which fulﬁlls Av = a ,

A: X → Y

is a given operator .

(6.1)

Let the numerical procedure constructed in order to solve this problem also be described in a general form by An vn = an ,

A n : X n → Yn , n ∈ N ,

(6.2)

where {an |an ∈ Yn , n ∈ N} is a given sequence approximating a ∈ Y in a sense that is still to be speciﬁed. Neither Xn ⊂ Xn+1 nor Xn ⊂ X and neither Yn ⊂ Yn+1 nor Yn ⊂ Y (n = 1, 2, . . .) are assumed to hold. Assume either that there are mappings pn : X → Xn , A X pn

q n : Y → Yn

- Y qn

?

An

Xn

? - Yn

or that there is an embedding of the numerical procedure, just for theoretical purposes, into the framework of the original problem. This can for instance be done in the sense of Xn ⊂ X ,

Yn = Y (n = 1, 2, . . .) ,

but not necessarily Xn ⊂ Xn+1

(n = 1, 2, . . .) .

Just for convenience, let us suppose that such an imbedding is realized, but the original problem does not necessarily need to be linear, and its solutions are allowed to be weak in a sense that is still to be speciﬁed. Let the original problem be described by ˜ =˜ Av a, ˜ X ˜ → Y, X ˜ ⊂ X, ˜ with A: a ∈ Y a given right hand side.

(6.3)

115

116

6 Proving the Existence of Entropy Solutions by Discretization Procedures

It is supposed that the weak formulation of (6.3), which will then be the actual problem to solve, is described in the following way. Let J be an index set, let {a(Φ), Φ ∈ J} ⊂ Y be a given set, and let {A(Φ), Φ ∈ J} be a set of operators with a joint domain D ⊂ X, D → Y. We ask for an element v ∈ X with A(Φ)v = a(Φ) ∀ Φ ∈ J .

(6.4)

Deﬁnition The elements of Θ := {v ∈ X|v solves (6.4)} are called weak solutions of (6.3) if the following implications hold: ˜ ∩ Θ, it also solves (6.3). If v solves (6.3), then v ∈ Θ; if v ∈ X The elements Φ ∈ J are called test elements. Also, let the numerical method be described in a rather general form by ˆ n vn = aˆn A

(n = 1, 2, . . .)

(6.5)

ˆ n : Xn → Z (n = 1, 2, . . .), with a certain topological space Z. Here, {ˆan } ⊂ Z where A is a given sequence that is compact in Z. Assume Θn := {vn ∈ Xn |vn solves (6.5)} =/ ∅ (n = 1, 2, . . .) , but that every Θn may contain more than one element. The solutions vn of (6.5) are called approximate solutions of (6.4). It is assumed that numerical method (6.5) does not depend on test elements, because computers cannot understand what test elements are. However, we also suppose that this method can be formulated weakly via a sequence of operators {An (Φ)|Φ ∈ J , n ∈ N} with An (Φ) : Xn → Yn , and a sequence {an (Φ)} with lim an (Φ) = a(Φ)

n→∞

for every ﬁxed Φ ∈ J ,

so that An (Φ)vn = an (Φ) ∀ Φ ∈ J , ∀ vn ∈ Θn holds. Equation (6.6) is called a weak formulation of the numerical method. Some concepts help to prove certain convergence results.

(6.6)

6.3 Convergence Theorems

Deﬁnition A pair [{Cn }, C ] consisting of an operator sequence {Cn } and an operator C is said to be asymptotically closed if the implication vn → v

∧

Cn vn → z

⇒

Cv = z

holds. Deﬁnition An operator sequence {Cn } is said to be asymptotically regular if the implication {Cn vn } compact in Y ⇒ {vn } compact in X holds. Deﬁnition Method (6.5) is called convergent if set convergence Θn → Θ is ensured in the following sense: {Θn } is discretely compact, i.e., every sequence {vn |vn ∈ Θn ; n = 1, 2, . . .} is compact in X. If v is a limit of a convergent subsequence, v ∈ Θ follows.

s

Remark Θn → Θ implies Θ ∈ / ∅. In other words, there is a solution of (6.4), i.e., a weak solution of (6.3). 101)

6.3 Convergence Theorems

We can now state a convergence theorem. Theorem 6.1 ˆ ˜ (i) Let {An (Φ)} , A(Φ) be asymptotically closed for every ﬁxed Φ ∈ J, and let n o ˆ n be asymptotically regular. (ii) A Then: Θn → Θ . 101) A proof of Θn → Θ is therefore also an existence proof.

117

118

6 Proving the Existence of Entropy Solutions by Discretization Procedures

Proof:

n o ˆ n vn |vn ∈ Θn compact in Z, and this holds for every {ˆan } compact in Z ⇒ A sequence {vn |vn ∈ Θn }. Thus, each of the sequences {vn } is compact in X because of assumption (ii), i.e.,

∃ {vn |vn ∈ Θn , n ∈ N ⊂ N} ,

∃v ∈ X with vn → v (n ∈ N ) .

Here, N and v are independent of Φ. However, because vn also fulﬁlls An (Φ)vn = an (Φ), and because of an (Φ) → a(Φ) ∀ Φ ∈ J , supposition (i) yields A(Φ)v = a(Φ), i.e., v ∈ Θ. If one is only interested in a particular problem, i.e., in a particular right hand side of equation (6.3), assumption (ii) can obviously be replaced by the weaker requirement (ii∗ )

{Θn } discretely compact.

If the uniqueness of a weak solution is not guaranteed, sometimes additional restrictions are formulated in order to select from among the set of weak solutions the particular one that is relevant from the user’s point of view. Also, as in the case of scalar conservation laws, this restriction often takes the form of an inequality that the relevant solution must additionally fulﬁll. Therefore, we will generally call such a restriction an entropy condition, and we formulate this condition in a general setting as ˆ u 0 ∀Φ ˆ ∈ ˆJ , B(Φ)v (6.7) n o ˆ Φ ˆ ∈ ˆJ denotes a set of nonlinear continuous functionals mapping X where B(Φ)| into R, and where ˆJ is a certain index set which may differ from J. We suppose that there is a uniqueness theorem available: 102) There is at most one solution of (6.4) that also fulﬁlls (6.7); this is termed the entropy solution. If the original problem is accompanied by an entropy condition in order to characterize the relevant solution, it seems reasonable to also discretize this condition by ˆ nu0 Bn (Φ)v

ˆ ∈ ˆJ (n = 1, 2, . . .) , ∀Φ

(6.8)

and to accept only approximate solutions vn ∈ Θn which additionally fulﬁll this ˆ are also assumed to be discrete entropy version. In this case, the functionals Bn (Φ) continuous. 102) Theorem 3.3 is an example.

6.3 Convergence Theorems

We suppose that {(6.5), (6.8)} has at least one solution vˆn ∈ Θn for every ﬁxed n ∈ N. (6.8) is called a sequence of discrete entropy conditions, but we do not expect a discrete entropy solution vˆn

(n ∈ N arbitrary)

to necessarily be unique. Deﬁnition An operator sequence {Cn } is said to be continuously convergent to an operator C if the implication vn → v

⇒

Cn vn → Cv

holds. 103)104) We then write c

Cn → C . We can now supplement Theorem 6.1 with another important statement: Theorem 6.2 Let ˆ n := {ˆ Θ vn ∈ Θn |ˆ vn solves (6.8)} =/ ∅ , and let c ˆ → ˆ be fulﬁlled for every ﬁxed Φ ˆ ∈ ˆJ . B(Φ) Bn (Φ)

Moreover, assume that the suppositions of Theorem 6.1 are fulﬁlled. Then there is an entropy solution vˆ of {(6.4),(6.7)}, and ˆ n → {ˆ Θ v} . Proof: Because of the validity of Theorem 6.1, i.e., because of Θn → Θ, there is a subsequence n o ˆ n , n ∈ N ⊂ N vˆn |ˆ vn ∈ Θ and a weak solution v ∈ Θ with {ˆ vn } → v, n ∈ N . 103) P. du Bois-Reymond: Sitz.-Berichte, Preussische Akad. d. Wiss. (1886) 359–360 104) R. Courant: Göttinger Nachr. (1914) 101–109

119

120

6 Proving the Existence of Entropy Solutions by Discretization Procedures

ˆ vn → B(Φ)v. ˆ The continuous convergence therefore yields Bn (Φ)ˆ ˆ u 0 follows. Consequently, v is an entropy solution, and beFrom (6.8), B(Φ)v cause the entropy solution was assumed to be unique, v coincides with this entropy solution, i.e., v = vˆ, and so both the subsequence and the full sequence converge to vˆ: lim vˆn = vˆ .

n→∞

The conditions allowing for the property of continuous convergence on metric spaces follow from a theorem provided by W. Rinow: 105)106) Theorem 6.3 If the operators C, Cn (n = 1, 2, . . .) mapping a metric space X into a metric space Y c are individually continuous, then Cn → C is equivalent to the simultaneous validity of the two statements: (i) (ii)

{Cn } is a sequence of equicontinuous operators ⊂ ˜ dense X Cn → C holds pointwise on a subset X

6.4 Example

As an example, we treat the scalar version of problem (2.1). Thus, let x ∈ R be a spatial variable and t v 0 a time variable, and let Ω again be the upper (x, t) half-plane. We ask for solutions v : Ω → R of the scalar conservation law ∂t v + ∂x f(v) = 0, t > 0 v(x, 0) = v0 (x) with v0 ∈ BV(R) ∩ L∞ (R) ,

(6.9)

where the ﬂux f ∈ C1 is strictly convex and has its minimum (assumed to be zero) at zero: f(0) = 0. Let ˘ ¯ ∗ | f |∞ := max | f (v)|, |v| u v0 L∞ < ∞ . (6.10) Because of possible discontinuities, we are interested in weak global solutions v ∈ X := Lloc 1 of Z Z {v∂t Φ + f(v)∂x Φ} dx dt A(Φ)v := – Ω

Z∞ – v0 (x)Φ(x, 0) dx = 0 –∞

105) W. Rinow: Die Innere Geometrie der Metrischen Räume. Berlin: Springer 1961, 78ff. 106) See also Section 7.9

(6.11) ∀ Φ ∈ J := C10 (Ω) .

6.4 Example

Let ˜ = S(v, ˜ c) , S

R~R→R

be a one-parameter set of so-called entropy functions that are continuously differentiable for every ﬁxed c ∈ R with respect to v, possibly with an exception at v = c. ˜ = F(v, ˜ c)(R ~ R → R) is called the entropy ﬂux if F ˜ ˜ ∂t S(v(x, t), c) + ∂x F(u(x, t), c) = 0 on Ω

holds for every ﬁxed c ∈ R and for every pair (v, Ω ) for which v is a weak solution on Ω but even a smooth C1 -solution on Ω ⊂ Ω. If a solution v which is weak on Ω additionally fulﬁlls the inequality ˜ ˜ t), c) + ∂x F(v(x, t), c) u 0 ∀ c ∈ R ∂t S(v(x, on Ω in a weak sense, i.e., Z Z ¯ ˘ ˜ ˜ – ∂t Φ(x, t)S(v(x, t), c) + ∂x Φ(x, t) F(v(x, t), c) dx dt Ω

Z∞

–

˜ 0 (x), c) dx u 0 Φ(x, 0)S(v

(6.12)

–∞

∀c ∈ R ,

∀ nonnegative Φ ∈ J ,

then v is called the entropy solution, denoted by vˆ. Thus, test elements in this example are the pairs ˆ := (Φ, c) with Φ ˆ ∈ ˆJ := {(Φ, c)|Φ ∈ J , Φ v 0 , c ∈ R} , Φ and the integral inequality can therefore be written as ˆ u 0 ∀Φ ˆ ∈ ˆJ B(Φ)v

(6.13)

˜ in the literature are in accordance with the general theory. Particular choices of S ˜=˜ S S(v) ,

˜ S

independent of c but strictly convex,

ˆ = {Φ ∈ J|Φ v 0} ⊂ J Φ (Lax; cf. (3.11)) , ˜ c) = |v – c| S(v, ( v – c, ˜ S(v, c) = 0 ,

(Kruzkov; cf. (3.44)) , vvc v
(Harten, Hyman, Lax) 107)

etc. ˜ according to (3.13). Each of these choices then implies a particular choice of F 107) A. Harten, J.M. Hyman, P.D. Lax: Comm. Pure Appl. Math. 29 (1976) 297–322

121

122

6 Proving the Existence of Entropy Solutions by Discretization Procedures

Let the numerical procedure used in the general setting be directed into a more concrete form using a (2k + 1)-point ﬁnite difference scheme: vjν+1 – vjν Δt

gνj+ 1 – gνj– 1 2

+

2

Δx

=0,

(6.14)

where ν counts the number of time steps and j the number of spatial steps, starting from vj0

1 = Δx

( j+1/2)Δx Z

v0 (x) dx . ( j–1/2)Δx

Here, the numerical ﬂux “ ” ν ν ν ν gνj+ 1 := g vj–k+1 , vj–k+2 , . . . , vjν , vj+1 , . . . , vj+k 2

is assumed to be Lipschitz continuous and consistent with the original problem, i.e., g(w, w, . . . , w) = f(w) , ∀ w ∈ R .

(6.15)

Methods of this form are called methods in conservation form. We restrict the choices of the step sizes by requiring that the CFL condition, introduced at the beginning of this chapter, must be respected. For the problem considered here, this means that Δt 1 = const u ∗ . Δx | f |∞

0 < λ :=

(6.16)

Let Δx = O( n1 ) (n ∈ N), and put (`

vn (x, t) =

vjν

for

j–

1 2

´

Δx u x

ν Δt u

t

` ´ < j + 12 Δx

< (ν + 1)Δt

( j = 0, ±1, ±2, . . .) , (ν = 1, 2, . . .) . If Xn is then chosen as the space of functions deﬁned on Ω and constant on these rectangles, the ﬁnite difference scheme can be formulated as ˆ n vn = 0 (n = 1, 2, . . .) A

(6.17)

ˆ n : Xn → Z := R. with operators A Multiplication of (6.14) by test functions Φ ∈ J and integration leads to An (Φ)vn =

(6.18) (ν+1)Δt ( j+ 12 )Δx

∞ X Z X ν=0

j

Z

h “ ”i Φ(x, t) vjν+1 – vjν + λ gνj+ 1 – gνj– 1 dx dt = 0 . 2

νΔt

( j– 12 )Δx

2

6.4 Example

This describes the operators An (Φ) as well as an (Φ) on the right hand side, which equal zero in this context. Using the abbreviation “ ” ˜νj (c) = ˜ S S vjν , c , vn is said to be a discrete entropy solution, denoted by vˆn , if ˜ν+1 (c) – S ˜ν (c) Gνj+ 1 (c) – Gνj– 1 (c) S j j 2 2 + u 0 ∀c ∈ R, Δt Δx where

(6.19)

ν ν ν , . . . , vjν , vj+1 , . . . , vj+k , c) Gνj+ 1 (c) := G(vj–k+1 2

denotes for every ﬁxed c a Lipschitz-continuous numerical entropy ﬂux that is still to ˜ in the sense of be speciﬁed and is consistent with the entropy ﬂux F 2 ˜ G(w, w, . . . , w, c) = F(w, c) ∀ (w, c) ∈ R . (6.20) For k = 1 (three-point case), the ﬂux splitting choice ˜– (β, c) ˜+ (α, c) + F G(α, β, c) = F with

(

˜+ (α, c) := F

˜ c) , F(α, 0 ,

αv0 , α<0

(

˜– (β, c) := F

(6.21) 0 , ˜ c) , F(β,

βv0 β<0

˜ is known as soon as S ˜ is chosen. is one of several possible choices of G, where F The weak formulation of the discrete form of the entropy condition needed to use Theorem 6.2 arises from the multiplication of (6.19) by nonnegative test functions Φ ∈ J and the following integration: ( Z Z ˜ 1 ˆ˜ ˜ n (x, t), c) ˆ Bn (Φ)vn := S(vn (x, t + Δt), c) – S(v Φ(x, t) Δt Ω

1 ˆ G(vn (x, t), vn (x + Δx, t), c) Δx ) ˜ dx dt u 0 , – G(vn (x – Δx, t), vn (x, t), c)

+

(6.22)

ˆ ∈ ˆJ , ∀Φ i.e., ˆ vn = – Bn (Φ)

Z∞ Z∞

Φ(x, t) – Φ(x, t – Δt) ˜ S(vn (x, t), c) dx dt Δt

Δt –∞ ZΔt

1 – Δt Z Z

– Ω

Φ(x +

0

xi +Δx/2 X Z ` ´ ˜ vi0 , c dt Φ(x, t) dx S

(6.23)

i x –Δx/2 i

Δx 2 , t)

– Φ(x – Δx

Δx 2 , t)

` ` G vn x –

´

Δx 2 ,t

` , vn x +

´ ´ , c dx dt

Δx 2 ,t

u0,

ˆ ∈ ˆJ . ∀Φ

123

124

6 Proving the Existence of Entropy Solutions by Discretization Procedures

Thus, B and Bn from (6.7) and (6.8), respectively, are placed in a concrete form provided that a three-point scheme is being considered. If, for instance, the Harten– ˜ is taken into account, and G is chosen according to (6.21), Hyman–Lax choice of S ˆ then B(Φ) in (6.12) and (6.13) is continuous for every ﬁxed Φ. The functionals Bn (Φ) (n = 1, 2, . . .) in (6.22) are equicontinuous, and the pointwise convergence ˆ → B(Φ) ˆ can easily be demonstrated provided that the joint domain of these Bn (Φ) functionals is restricted to elements v ∈ X∗ ; here, X∗ is the space of functions v ∈ X restricted to the domain supp(Φ), which is equipped with the L1 metric on X∗ . Hence, the assumptions (i) and (ii) of Rinow’s theorem are fulﬁlled, i.e., c

Bn (Φ) → B(Φ) , ˆ n =/ ∅. 108) and Theorem 6.2 applies as far as Θ 109) The explicit Engquist–Osher three-point ﬂux splitting scheme, 110) characterized by (6.14) with “ ” “ ” “ ” ν ν gνj+ 1 = g vjν , vj+1 := f– vj+1 + f+ vjν (6.24) 2

where (

f+ (α) =

f (α) 0

for for

αv0 , α<0

(

f– (α) =

0 f (α)

for for

αv0 , α<0

˜ is an example. (6.24) then detogether with the Harten–Hyman–Lax choice of S, 111) ˆ scribes the operators An (n = 1, 2, . . .) in (6.17), and hence also the operators An (Φ) via (6.18). The method simulates an important property of weak solutions of scalar problems of type (2.1) provided the CFL condition is fulﬁlled, namely their monotonicity, and hence their total variation diminishing (TVD) nature and thus their total variation boundedness (TVB). We interrupt the proof of our convergence theorem for some explanations. A method used for the numerical treatment of scalar conservation laws of type (2.1), represented in the general form “ ” n n n vjn+1 = H vj–k , vj–k+1 , . . . , vj+k , is said to be monotone if the partial derivatives of the function H with respect to all of its variables are nonnegative. This imitates the following property of the weak solutions of (2.1). If two initial functions v˜0 and v0 fulﬁll the inequality v˜0 (x) v v0 (x)

∀x ∈ R ,

this inequality holds everywhere in Ω for the weak solutions v˜(x, t) and v(x, t) belonging to these initial functions, respectively. 108) See footnote 97 109) It follows from this explicitness that Θn =/ ∅ (n = 1, 2, . . .).

110) B. Engquist, S. Osher: Math. Comput. 34 (1980) 45–75 111) aˆn = 0 (n = 1, 2, . . .)

6.4 Example

In this context (and formulated for systems as well as scalar equations), TVP n boundedness means that j Vj+1 – Vjn Rm u K for all n, and TVD methods intro112) duced by Harten simulate the fact that the total variation of weak solutions of scalar problems does not increase, and it is obvious that TVD implies TVB. The fact that monotonicity of a scheme implies that it has TVD properties was shown in an appendix to the paper of Harten, Hyman and Lax (cf. footnote 107), written by Barbara Keyﬁtz. We now resume our proof of the convergence theorem. LeVeque 113) showed that Θn is discretely compact 114) provided that v0 shows bounded variation. It is here that we put our requirement v0 ∈ BV(R) to use. Thus, assumption (ii) in Theorem 6.1 is fulﬁlled because (ii∗ ) is fulﬁlled. Property (i), i.e., [{An (Φ)}, A(Φ)] discretely closed , remains to be shown in order to demonstrate the convergence of the method using Theorem 6.1. However, in the case of TVB methods for scalar equations, this can immediately be deduced from the Lax–Wendroff theorem. 115) This ends the proof for the existence of an entropy solution to our scalar conservation law example. As a particular result, we note that a convergent monotonic consistent scheme in conservation form converges to the entropy solution of the original problem. The Lax–Wendroff theorem referred to in the proof of the convergence theorem reads as follows. Theorem 6.4 Δt Let λ = Δx = const and let Δt = O the step function

V n (x, t) := Vjn for xj – ( j = 0, ±1, ±2, . . . ;

`1´ n

(n ∈ N) for n → ∞. Deﬁne V n ∈ Lloc 1 to be

Δx Δx u x < xj + , 2 2

tn u t < tn+1 ,

n = 0, 1, 2, . . .) on Ω = {(x, t) | x ∈ R , t v 0} ,

where the numerical values Vnj were computed via an explicit TV-bounding ﬁnite difference one-step scheme whose numerical ﬂux g is consistent and Lipschitzcontinuous, and which starts from 1 V0j := Δx

xj + Δx 2

Z

V0 (ξ) dξ ( j = 0, ±1, ±2, . . .) . xj – Δx 2

112) A. Harten: SIAM J. Numer. Anal. 21 (1984) 1–23 113) R.J. LeVeque: Numerical methods for conservation laws. Basel: Birkhäuser 1990, p. 164 114) However, LeVeque used another terminology.

115) P.D. Lax, B. Wendroff: Comm. Pure Appl. Math. 13 (1960) 217–237

125

126

6 Proving the Existence of Entropy Solutions by Discretization Procedures

Assume that V n converges for n → ∞, i.e., for Δt → 0, to a function V in the i.e., with respect to the L1 -norm on each compact subset of Ω and for each of the components of V n . Then V is a weak solution of (2.1).

Lloc 1 -topology,

We omit the proof of the Lax–Wendroff theorem, but note that this is not a convergence theorem yet! The theorem only says that the limits of convergent sequences of approximate solutions – assuming they exist – are weak solutions of the original problem. In order to complete a proof that demonstrates the important property that a method converges, one must look for properties which ensure the validity of the assumptions of Theorems 6.1 and 6.2 (as was done in the case of the Engquist– Osher scheme). It should be mentioned that obviously the general convergence theorem of this chapter applies not only to initial value problems for scalar conservation laws, but also to certain nonlinear elliptic boundary value problems that are numerically treated using projection methods, etc. 116)

116) R. Ansorge, J. Lei: ZAMM 71 (1991) 207–221

127

7 Types of Discretization Principles 7.1 Some General Remarks

In this chapter we are going to describe the principal ways of discretizing hyperbolic conservation laws, as well as their common features and differences between them. 117) For the sake of simplicity, we consider the generic scalar equation L(u) := ∂t u + ∂x u + ∂y u = 0 .

(7.1)

There are three basic classes of discretization method for (7.1), namely the ﬁnite element method (FEM), the ﬁnite volume method (FVM), and the ﬁnite difference method (FDM), already introduced brieﬂy in Section 4.1 and used for theoretical purposes in Section 6.3. Although these names sound quite deﬁnitive, there are great many particular schemes in each of the three method classes. In the FDM, we begin by discretizing time Gn := {nΔt | Δt > 0 , n ∈ N ∪ {0}} as well as space Gi,j := {(iΔx, jΔy) | Δx, Δy > 0 , i, j ∈ Z} and replace the differential operators in (7.1) by difference operators, for example: ∂t u(iΔx, jΔy, nΔt) W =:

u(iΔx, jΔy, (n + 1)Δt) – u(iΔx, jΔy, nΔt) Δt n un+1 i,j – ui,j Δt

and ∂x u(iΔx, jΔy, nΔt) W =: 117) See also Chapter 6

u(iΔx, jΔy, nΔt) – u((i – 1)Δx, jΔy, nΔt) Δx uni,j – uni–1,j Δx

128

7 Types of Discretization Principles

u(iΔx, jΔy, nΔt) – u(iΔx, (j – 1)Δy, nΔt) Δy uni,j – uni,j–1 . =: Δy

∂y u(iΔx, jΔy, nΔt) W

Replacing the derivatives in (7.1) by the differences leads to the difference equation Lh (u) :=

n un+1 i,j – ui,j

Δt

+

uni,j – uni–1,j Δx

+

uni,j – uni,j–1 Δy

W0,

where the superscript h indicates a discrete value (it is very common to encounter Δx = Δy and simply call this value h). Since we decided to approximate the spatial derivatives at time step nΔt, the resulting difference equation is explicit, i.e., the solution at the resulting time step (n + 1)Δt is explicitly computable through n un+1 i,j W ui,j –

” Δt “ ” Δt “ n ui,j – uni–1,j – uni,j – uni,j–1 . Δx Δy

(7.2)

If we had decided to take the spatial differences at t = (n + 1)Δt, we would have ended up with an implicit equation, requiring the solution of a system of linear equations at each time step. Note that in (7.2) we still used the exact solution u of (7.1). Since there will be errors due to the replacement of differential quotients by difference quotients, we cannot assume that (7.2) will hold for u with equals sign. Thus, (7.2) should be rewritten as n Un+1 i,j = Ui,j –

” Δt “ ” Δt “ n Ui,j – Uni–1,j – Uni,j – Uni,j–1 , Δx Δy

(7.3)

and (7.3) now deﬁnes an approximation of Uni,j to uni,j within the bounds of the discretization and other errors. Also note that the numerical solution is represented pointwise within the FDM framework, i.e., we know pointwise values of approximations of the exact solution. We conclude with a philosophical remark: In FDMs, one discretizes the operators that occur in the differential equation. The discrete solution Uh := (Uni,j |i, j ∈ Z n ∈ N) consists of point values at the grid points of Gn ~ Gi,j . In the FEM, one always starts from a weak (variational) formulation like (2.2). The space Ω := R2 ~ {t ∈ R | t v 0} is divided into certain elements using a grid. The chosen elements may be the space-time bricks used to build the ﬁnite difference grid Gn ~ Gi,j , but they can also be far more general. Due to their simplicity and their ability to resolve even complex geometries, triangulations are often used when the elements are triangles or tetrahedra. Applying the notion of weak solutions and choosing a space W of test functions leads to Z ˘ ¯ ∀Φ ∈ W : u∂t Φ + u∂x Φ + u∂y Φ dt dx dy = 0 . Ω

7.1 Some General Remarks

If the test function space is replaced by a ﬁnite-dimensional subspace Wh , the relation Z n o u∂t Φh + u∂x Φh + u∂y Φh dt dx dy = 0 . ∀ Φh ∈ Wh : Ω

must be fulﬁlled. This is a set of as many equations as the dimension of Wh := span{Φ1 , . . . , Φn }. A popular choice is the space of polynomials linear in t and with x, y vanishing outside each element Σ ⊂ Ω. The choice of U(x, y, t) :=

n X

Ui Φi (x, y, t)

i=1

as an approximation for u then leads to a linear system for the unknown coefﬁcients Ui since Z ˘ ¯ U∂t Φj + U∂x Φj + U∂y Φj dt dx Ω

=

Z (X n

=

U i Φi ∂ t Φj +

i=1

Ω n X

n X

U i Φi ∂ x Φj +

i=1

n X

)

U i Φi ∂ y Φj

dt dx dy

i=1

0 1 0 1 Z Z n X Ui @ Φi ∂t Φj dt dx dyA + Ui @ Φi ∂x Φj dt dx dyA

i=1

i=1

Ω

Ω

0 1 Z n X + Ui @ Φi ∂y Φj dt dx dyA i=1

Ω

0 1 Z n X @ Φi (∂t Φj + ∂x Φj + ∂y Φj ) dt dx dyA Ui = 0 , = i=1

Ω

(∀ j = 1, . . . , n) . FEMs of the type just described are usually called space-time FEMs. It is often convenient to only discretize in space with ﬁnite elements and tackle the time derivative in a ﬁnite difference manner. In order ro remove the coupling of every element with any other element, one can use discontinuous Galerkin methods, where discontinuities are allowed at element faces. This type of method is very similar to ﬁnite volume methods, but not identical. Also popular are streamline diffusion FEMs, which belong to the class of Petrov–Galerkin schemes. Petrov–Galerkin methods are characterized by the selection of a test space that is different from the Ansatz space in which the discrete solution is sought. Although some mathematical analysis is available for streamline diffusion methods, the numerical results obtained so far cannot cope with the results obtained by FDMs.

129

130

7 Types of Discretization Principles

Again we end up with a philosophical remark: In FEMs, one discretizes the solution of the differential equation. The discrete solution consists of polynomials deﬁned piecewise on the elements. The third class of methods, FVMs, share all of the good properties of FDMs as well as of those FEMs. As with FEMs, arbitrary cells in space can be used for the discretization, while all of the FD technology is also applicable at the same time. Here, one starts with a type of weak solution which stems directly from gas dynamics, where the conservative equations are formulated as integral balances on so-called control volumes. For our generic model, this weak form results from integrating ∂t u + ∂x u + ∂y u = ∂t u + ∇, u = 0 ,

u := (u, u)T

over a bounded control volume σ ⊂ R2 with boundary ∂σ, where the unit outer normal vector n is deﬁned almost everywhere. From Gauss’ divergence theorem it then follows Z I d u dx dy + u, n ds = 0 . dt σ

∂σ

Now, the cell average of u on σ is deﬁned as Z 1 uσ (t) := u(x, y, t) dx dy , |σ| σ

where |σ| denotes the area of σ, so that the weak formulation of gas dynamics unfolds itself as an evolution equation for cell averages I 1 d uσ = – u, n ds . (7.4) dt |σ| ∂σ

FVMs are derived from (7.4) by approximating the line integrals with a suitable Gauss formula. The product u, n is replaced by a so-called numerical ﬂux function which is essentially a ﬁnite difference formula. Again we conclude with a philosophical remark: FVMs are discrete evolution equations for cell averages. There is freedom in relation to representing the numerical solution. Some people like to view FVMs as being part of the Petrov–Galerkin family of FEMs with piecewise constant trial functions. The central idea in FVMs is to adapt FDMs to unstructured grids like triangulations and to use the computational advantages from FDMs and FEMs. There are even more discretization methods available than the three classes described here, but they are often classiﬁed as particular instances of FDMs, FEMs and FVMs. A method that is popularly used in the computation of incompressible

7.2 Finite Difference Calculus

viscous ﬂows, and is now becoming more and more attractive for compressible ﬂuid ﬂow, is the class of spectral discretizations. Here the Ansatz space consists of globally deﬁned trigonometric or polynomial functions on the whole domain (or, as in the spectral element method, on macroelements). Choosing an FEM-like Ansatz leads to spectral Galerkin methods. If the differential equation must be satisﬁed pointwise at so-called collocation points, one talks about spectral collocation or pseudospectral methods. These methods are very close to being FDMs with large stencils.

7.2 Finite Difference Calculus

Let us now study the structures of ﬁnite difference methods more intensively. A century before the invention of calculus by Leibniz and Newton, the foundations had been laid for a discrete calculus employing differences in functions. Napier’s 118) and Briggs’ 119) invention of logarithms revolutionized naval navigation and necessitated interpolation formulae for computing intermediate values for tabulated data. It was the genius Thomas Harriot 120) who founded the ﬁeld of ﬁnite difference calculus. This ﬁeld ﬂourished in the days of Leibniz and Euler and came to exert enormous inﬂuence in the area of the numerical treatment of ordinary and partial differential equations. To again give the reader an idea of the spirit of ﬁnite difference calculus, we consider the forward difference operator Δ deﬁned by Δu(x) := u(x + Δx) – u(x) , where Δx denotes an increment that is different from zero. It is often convenient to introduce the shift operator S through Su(x) := u(x + Δx) so that Δ = S – I, where I is the identity operator. Assuming a smooth function x → u(x), Taylor series expansion results in u(x + Δx) =

∞ X 1 ν ν Δx D u(x) , ν! ν=0

where Dν is deﬁned as

d

|dx

◦

d dx

d ◦ . . . ◦ dx and D0 = I. {z }

ν times

Utilizing the shift operator, this is simply the representation ! ∞ X 1 ν ν Su(x) = Δx D u(x) , ν! ν=0 118) John Napier (1559–1617); Merchiston 119) Henry Briggs (1556–1630); London, Oxford 120) Thomas Harriot (1560–1621); Oxford

131

132

7 Types of Discretization Principles

where now, formally, the term in parentheses is exactly eΔxD , i.e., we have found the representation S = eΔxD

(7.5)

and hence Δ = eΔxD – I .

(7.6)

Introducing the central difference operator δ by means of δu(x) := u(x + Δx/2) – u(x – Δx/2) , a further Taylor series u(x ± Δx/2) =

∞ X 1 1 (±Δx)ν ν Dν u(x) ν! 2 ν=0

reveals δu(x) = 2

∞ X ν=0

1 (2ν + 1)!

„

Δx D 2

«2ν+1

u(x) ,

i.e., „

δ = 2 sinh

Δx D 2

«

.

(7.7)

Inverting this equation gives a relation between the differential operator D and the central difference operator δ, namely D=

δ 1 2 sinh–1 = Δx 2 Δx

„

δ–

« 12 3 12 32 5 + – . . . δ δ 22 3! 24 5!

(7.8)

where δn := |δ ◦ δ ◦{z. . . ◦ δ}. Equation (7.8) is an exact difference formula for the opern times

ator D. Other equations like (7.8) and (7.6) can also be constructed. We are mainly interested in the applicability of formulae like (7.8). To get an idea, we consider the simple linear transport equation ∂t u + ∂x u = 0, which we rewrite as ∂t u = L(D)u := –∂x u , where the differential operator D is now a partial derivation with respect to x. From (7.5), we deduce “ ” u(x, t + Δt) = eΔt∂t u(x, t) and, since ∂t = L(D), we conclude “ ” = eΔtL(D) uni . un+1 i

(7.9)

7.2 Finite Difference Calculus

We now use (7.8) to eliminate D in terms of δ in (7.9). This gives “ ” –1 un+1 = eΔtL(2/Δx sinh (δ/2)) uni . i

(7.10)

Note that (7.10) is not an approximate difference equation but an exact one. Retaining only the linear terms in the exponential function gives “ ” Un+1 = I + ΔtL(2/Δx sinh–1 (δ/2)) Uni . i Retaining only linear terms in the expansion (7.8) and noting that L(D) = –D = –∂x ﬁnally leads to „ « δ = I – Δt Uni . Un+1 i Δx Expanding the central difference operator gives the ﬁnite difference method = Uni – Un+1 i

” Δt “ n Ui+1/2 – Uni–1/2 Δx

(7.11)

for the transport equation. Hence, ﬁnite difference calculus can be used directly to construct ﬁnite difference methods for partial differential equations. More constructions of this type can be found in the literature. Now let the above difference scheme act on the true solution u. Again employing Taylor series expansions, un+1 = u(x, t) + Δt∂t u(x, t) + O(Δt2 ) i Δx 1 Δx2 2 ∂ u(x, t) + O(Δx3 ) ∂x u(x, t) + 2 2! 22 x Δx 1 Δx2 2 = u(x, t) – ∂ u(x, t) + O(Δx3 ) , ∂x u(x, t) + 2 2! 22 x

uni+1/2 = u(x, t) + uni–1/2

and inserting them into (7.11) gives u + Δt∂t u + O(Δt2 ) = u –

´ Δt ` Δx∂x u + O(Δx3 ) Δx

which, after rearranging, leads to ∂t u + ∂x u = O(Δt, Δx2 ) .

(7.12)

Thus, our simple difference method (7.11) is of order one in time and of order two in space. This notion of order is usually called the truncation error since it results from truncating a Taylor series. Note that we have not said a word about the convergence of (7.11). The property that (7.12) formally tends to ∂t u + ∂x u = 0 for Δt, Δx → 0 is called consistency. ` ´ If a difference operator is applied to a sequence Uni |i ∈ Z , then it acts on the sequence index, i.e., Δi Uni := ΔUni = Uni+1 – Uni .

133

134

7 Types of Discretization Principles

For later reference, we now introduce the antidifference operator Δ–1 as follows. If Δui = gi holds, then we deﬁne ui = Δ–1 gi + c , where c denotes an arbitrary constant. The constant appears because after the application of Δ we get Δui = gi back. We ﬁrst prove n–1 X

Δui = un – un0 ,

(7.13)

i=n0

0 1 n–1 X Δ@ ui A = un .

(7.14)

i=n0

Equation (7.13) simply follows from n–1 X

Δui = (un0 +1 – un0 ) + (un0 +2 – un0 +1 ) + . . . + (un–1 – un–2 ) + (un – un–1 ) ,

i=n0

while (7.14) is simply 0 1 n–1 n–1 n–1 n–1 n–1 X X X X X ui A = ui+1 – ui = ui+1 – ui . Δ@ i=n0

i+1=n0

i=n0

i=n0 –1

i=n0

Using formula (7.14), we can immediately deduce Δ–1 un =

n–1 X

ui + c ,

(7.15)

i=n0

where n0 is an arbitrary start index. We now provide a discrete analog of the product rule, namely Δ(ui vi ) = Sui Δvi + vi Δui .

(7.16)

The proof consists of evaluating the right and left hand sides to see that they coincide. Writing the product rule in the form vi Δui = Δ(ui vi ) – ui+1 Δvi and applying the operator Δ–1 results in Δ–1 (vi Δui ) = Δ–1 Δ(ui vi ) – Δ–1 (ui+1 Δvi ). Following formula (7.15), we ﬁnally get the formula for summation by parts, which is a discrete analog to the formula of integration by parts, namely n–1 X i=n0

vi Δui = un vn –

n–1 X i=n0

ui+1 Δvi + c .

(7.17)

7.3 The CFL Condition

7.3 The CFL Condition

In 1928, Courant, 121) Friedrichs, 122) and Hans Lewy published a milestone paper (cf. Section 6.1) which was ignored during the war and only became famous when it was rediscovered during the Manhattan Project. As already emphasized in the last chapter, Courant’s interests at that time mainly involved proving the existence of solutions to partial differential equations. In that ﬁeld he was one of the founders of the idea of employing ﬁnite difference schemes as approximations to partial differential equations. Proving the existence of solutions to partial differential equations involved proving that the difference scheme was consistent and convergent, and furthermore that the limit solution (as Δt and Δx tend to zero) satisﬁed the original differential equation. Although Courant, Friedrichs and Lewy succeeded with this scheme for the (elliptic) Laplace equation, they encountered serious problems when applying it to the (hyperbolic) wave equation. They ﬁnally succeeded in formulating the celebrated CFL condition, which is necessary for convergence. In order to demonstrate the basic behavior observed in the context considered here, let us consider the simple forward explicit difference scheme for the slightly more general transport equation ∂t u + a∂x u = 0, a < 0, namely = Uni – a Un+1 i

´ Δt ` n Ui+1 – Uni = Δx

„ « Δt Δt 1+a Uni – a Uni+1 . Δx Δx

(7.18)

(7.18) is an explicit three-point scheme for the explicit computation of Un+1 i . Viewed in the (x, t)-plane, we can identify the difference stencil of (7.18), as shown in Fig. 7.1. The stencil gives rise to the deﬁnition of numerical characteristics, which are straight lines enveloping the stencil. The numerical characteristics cut off an interval of the axis t = 0, which is called the numerical domain of dependence. Changing the initial datum in this interval changes the numerical solution at point P, while perturbations outside this interval would not affect the numerical solution at P. However, there is also the characteristic Γ of the differential equation that we would like to approximate. If Γ lies outside the numerical characteristics, as shown in Fig. 5.8, then changing the initial datum between Γ and the numerical characteristic would change the true solution at P but not the numerical solution. Thus, convergence of the difference scheme is not possible. Keeping Γ inside the numerical characteristics ensures that all perturbations of the initial datum which affect the true solution at P will always affect the numerical solution at P too. The characteristic Γ has the slope dt 1 = dx a 121) Richard Courant (1888–1972); Göttingen, New York 122) Kurt Otto Friedrichs (1901–1982); Brunswick, New York

135

136

7 Types of Discretization Principles

Fig. 7.1 Difference stencil and characteristics.

and the right numerical characteristic has a slope of –Δt/Δx. Thus, Δt/Δx u 1/|a| is necessary for convergence, which is already the Courant–Friedrichs–Lewy (CFL) condition Δx Δt u . (7.19) |a| In the case of a > 0, and replacing the forward difference equation by the backward one, (7.19) again ensures appropriate behavior for the domains of dependence. This dependence of the type of difference equation on the sign of a, i.e., on the direction of movement of the wave, leads to so-called upwind schemes. In the case of a scalar conservation law ∂t u+∂x f (u) = 0, the same arguments hold for the quasilinear form ∂t u + f (u)∂x u = 0. Hence, instead of (7.19), explicit ﬁnite difference schemes with a three-point stencil have to satisfy the CFL condition Δt max | f (Uni )| u 1 Δx i

(7.20)

at each time step n.

7.4 Lax–Richtmyer Theory

We now formalize the treatment of ﬁnite difference schemes, supplementing Chapter 6, in order to present at least the basic historical idea of a convergence theory for this particular type of scheme. Although we have to conﬁne ourselves to linear problems, the Lax–Richtmyer theory allows us to gain deep insights into the interplay between the notions of consistency, stability and convergence. It was originally formulated for autonomous,

7.4 Lax–Richtmyer Theory

linear initial value problems, but was later generalized by several authors to include nonautonomous and nonlinear problems. 123) We consider an abstract evolution equation in a Banach space B, where a linear operator A: DA → B acts upon a subspace DA of B. The evolution equation we want to discretize is then given as d u(t) = A(u(t)), t ∈ [0, T ] , T > 0 dt u(0) = u0 , u0 ∈ DA .

(7.21)

A one-parameter family u(t) with t ∈ [0, T ] , u(t) ∈ DA , is called a genuine solution of (7.21), if ‚ ‚ ‚ u(t + Δt) – u(t) ‚ ‚ (7.22) – Au(t)‚ lim ‚ ‚ =0. Δt→0 Δt B 0utuT Let D be the set of all initial data u0 ∈ B so that a unique genuine solution exists with u(0) = u0 and such that convergence in (7.22) is uniform in t. 124) There is then a one-parameter family of linear operators E0 (t): D → B deﬁned by u(t) = E0 (t)u0 which are called evolution operators or solution operators. We always assume that our problem is properly posed, which in particular implies that E0 (t) is bounded uniformly in t, i.e., there is a constant K > 0 such that E0 (t)D→B < K for 0 u t u T. Then the Hahn–Banach extension theorem implies the existence of a generalized solution operator E(t): B → B with the same bound K as E0 (t). The generalized solution operator has the semigroup property ∀ s, t v 0 :

E(s + t) = E(s) E(t)

and the closure of the operator A, which we also denote by A, is the inﬁnitesimal generator of this semigroup. One can further show that E(t) commutes with A, so that E(t)Au0 = AE(t)u0 . Then (7.21) can be written as ‚ „ « ‚ ‚ ‚ E(Δt) – I E(t) u0 ‚ – A lim ‚ ‚ =0, Δt→0 ‚ Δt B 0utuT which, since E(t)B < K for 0 u t u T, implies uniform convergence of (7.22) with respect to t. 123) See R. Ansorge: Survey of equivalence theorems in the theory of difference approximations for partial initial value problems. In: J.J. Miller (ed.): Topics in

numerical analysis. London: Academic Press 1977 124) We shall see shortly that uniform convergence is automatic.

137

138

7 Types of Discretization Principles

In order to discuss fairly general ﬁnite difference schemes, let us introduce linear difference operators Λ0/1 = Λ0/1 (Δt, Δx1 , . . . , Δxd ) and the class of difference schemes Λ1 Un+1 = Λ0 Un . –1 We assume the existence of Λ–1 1 and that Λ1 Λ0 is a bounded linear operator deﬁned on B. From the CFL condition, we know that Δt and Δxi cannot go to zero independently of each other as far as explicit methods are concerned. Hence, we assume the relation Δxi = ϕi (Δt), i = 1, . . . , d and introduce the discrete evolution operator ` ´ ` ´ C(Δt) := Λ–1 1 Δt, ϕ1 (Δt), . . . , ϕd (Δt) · Λ0 Δt, ϕ1 (Δt), . . . , ϕd (Δt)

so that the class of difference schemes under consideration is Un+1 = C(Δt)Un .

(7.23)

The family of operators C(Δt) is said to deﬁne a consistent approximation of the initial value problem if, for every u(t) in a class of genuine solutions arising from initial data u(0) out of a set that is dense in B, the consistency condition ‚„ ‚ « ‚ C(Δt) – I ‚ ‚ (7.24) – A u(t)‚ lim ‚ ‚ =0 Δt→0 Δt B 0utuT holds. Since u(t) denotes a genuine solution, we can use the fact that, due to (7.22), Au(t) is close to (u(t + Δt) – u(t))/Δt in order to get the alternative deﬁnition ‚ ‚ ‚ u(t + Δt) – C(Δt)u(t) ‚ ‚ =0. (7.25) lim ‚ ‚ Δt→0 ‚ Δt B 0utuT The quantity ‚ ‚ ‚ u(t + Δt) – C(Δt)u(t) ‚ ‚ := ˆε(Δt, u0 ) supt∈[0,T ] ‚ ‚ ‚ Δt B

is called truncation error, and if this error is of order p, i.e., ˆε(Δt, u0 ) = O(Δtp ) , the method is called a method of order p. Consistency therefore means ˆε(Δt, u0 ) = o(1) which is fulﬁlled in particular for p > 0. Note that this abstract notion is in complete agreement with our earlier calculations in (7.12).

7.4 Lax–Richtmyer Theory

Consistency guarantees that, in the limit Δt → 0, the difference scheme formally reproduces the differential equation. The notion of convergence is concerned with the limit behavior of the discrete solutions. Consistency does not necessarily already imply convergence! Due to (7.23), the iterated application of C(Δt) gives Un = C(Δt)Un–1 = C(Δt) ◦ C(Δt)Un–2 = . . . = Cn (Δt)U0 = Cn (Δt)u0 , and this discrete function is expected to be a good approximation to u(nΔt) = E(nΔt)u0 . The family of operators C(Δt) provides a convergent approximation to the initial value problem if, for ﬁxed t ∈ [0, T ], ‚ ‚ lim ‚C(Δj t)nj u0 – E(t)u0 ‚B = 0 (7.26) j→∞

holds for each sequence (Δj t)j=1,2,... tending to zero, where nj is chosen to satisfy limj→∞ nj Δj t = t. In reaching the discrete solution at time t = nΔt, the operator C(Δt) has to be applied n times on the initial datum. If we consider small errors in u0 or errors due to rounding during the computation, we would not be satisﬁed with a difference scheme which would amplify these small perturbations until they spoil the solution. The central idea of numerical stability is therefore to put bounds on the possible ampliﬁcation of initial data. The operator C(Δt) is said to deﬁne a stable approximation if, for some t∗ > 0, the inﬁnite set of operators C(Δt)n ,

0 < Δt < t∗ ,

0 u nΔt u T ,

are uniformly bounded, i.e., if there is a constant κ with C(Δt)n B u κ , ∀ n ∈ N , ∀ Δt with 0 < Δt u t∗

and with 0 u nΔt u T .

With the notions of consistency, stability and convergence, we can now formulate the central equivalence theorem of the Lax–Richtmyer theory. Theorem 7.1 A ﬁnite difference scheme approximating a properly posed initial value problem is convergent if and only if it is consistent and stable. Proof (of the direction: consistency ∧ stability ⇒ convergence 125)): U(0) = u0 implies Un – u(t) =

n X

C(Δt)ν–1 {C(Δt) u (tn–ν ) – u (tn–ν+1 )} + u(tn ) – u(t)

ν=1

(tμ = μ Δt) . 125) From an engineering point of view, this is the more important direction. The complete proof can be found in the original paper of Lax and Richtmyer cited in footnote 99. If consistency and stability

are deﬁned properly, theorems of the Lax–Richtmyer type can even be proven in the case of nonlinear difference schemes for nonlinear equations, as previously mentioned in footnote 123.

139

140

7 Types of Discretization Principles

Thus, from the stability property, we obtain Un – u(t)B u

n X ‚ ‚ ‚C(Δt)ν–1 ‚ · C(Δt) u (tn–ν ) – u (tn–ν+1 ) B B ν=1

+ u(tn ) – u(t)B uκ

n X

C(Δt) E0 (tn–ν ) u0 – E0 (tn–ν+1 ) u0 B

ν=1

+ u(tn ) – u(t)B , leading to Un – u(t)B u κ n Δt · ˆε(Δt, u0 ) + u(tn ) – u(t)B . Taking n Δt u T into account, we ﬁnd Un – u(t)B u κ T εˆ(Δt, u0 ) + u(n Δt) – u(t)B . Because of n Δt → t, because of the continuity of u with respect to t, and because of the consistency condition (7.25), the truth of the assertion is obvious. Particularly for n Δt = t, we get an estimate for the global discretization error Un – u(t)B , namely Un – u(t)B u κ t ˆε(Δt, u0 ) . Round-off errors are not included here! Example Let us again treat the transport equation ∂t u + a∂x u = 0 in the case of a < 0. Thus, if (7.18) is expressed in terms of the Lax–Richtmyer theory, and if a relation between the step sizes Δt and Δx is described, we ﬁnd ˛ ˛ ˛ Δt ˛˛ Δt ˛ C(Δt)∞ u ˛1 + a + |a| . Δx ˛ Δx The CFL condition (7.19) then ensures that „ « Δt Δt + |a| =1, C(Δt)∞ u 1 + a Δx Δx and hence Cn (Δt)∞ u C(Δt)n∞ u 1 . Thus, the CFL condition guarantees the numerical stability of the scheme in the sense of the Lax–Richtmyer theory.

7.5 The von Neumann Stability Criterion

The other direction is also true: stability in the sense of the Lax–Richtmyer theory requires the CFL condition to be fulﬁlled, and this holds independently of the norm. Of course, the upwind scheme for a > 0 can be treated in the same way. The upwind idea can also be looked at from the point of view of information theory: looking in the direction of information travel, it seems unreasonable to use values for the present time step whose positions lie downstream and have not yet been inﬂuenced by the information used to compute a future value of the solution at a given position in space.

7.5 The von Neumann Stability Criterion

The consistency of a ﬁnite difference scheme is easy to verify via Taylor expansions. In contrast, it may be much harder to verify stability using the pure deﬁnition. In the case of linear initial value problems or problems with periodic boundary data, there is a simple mechanism that was derived by von Neumann. 126) We start with a complex Fourier transform of a discrete function represented as a sequence Un = (Unj )j∈Z and deﬁned at grid points only. The discrete Fourier transform is deﬁned as cn (ξ) = U

X

Unj e– ijξ ,

0 u ξ < 2π .

j∈Z

The importance of the discrete Fourier transform in analyzing the stability of a ﬁnite difference scheme can be seen if it is applied to the shift operator S deﬁned in Section 7.2. For our purposes, we reﬁne the deﬁnition by calling S+ Un := S(Δx)Un := (Unj+1 )j∈Z the forward shift operator. Therefore, it follows from the deﬁnition of the discrete Fourier transform that X X n S Unj+1 e– ijξ = Unj e– i(j–1)ξ + U (ξ) = j∈Z

=

X

j∈Z

Unj e– ij e iξ

j∈Z

= e iξ

X

Unj e– ijξ

j∈Z

iξ cn

= e U (ξ) . Thus, applying the shift operator to a grid function is, in Fourier space, multiplication with e iξ . The function e iξ is called the symbol of the shift operator S. Completely analogously, one obtains the symbol e– iξ for the backward shift operator S– Un := S(–Δx)Un , i.e., – iξ cn n S – U (ξ) = e U (ξ) .

126) John von Neumann (1903–1957); Berlin, Hamburg, Göttingen, Princeton

141

142

7 Types of Discretization Principles

Since every linear ﬁnite difference scheme is made up of combinations of forward and backward shift operators, we now have a way to look at discrete Fourier transforms of difference schemes. Example Again consider the difference scheme = Uni – Un+1 i

´ Δt ` n a Ui+1 – Uni–1 2Δx

as an approximation to ∂t u + a∂x u = 0. It is an easy exercise to see that the scheme is second-order consistent in space and ﬁrst-order consistent in time. Written in terms of shift operators and sequences, we get „ « Δt n+1 U = I– a(S+ – S– ) Un . 2Δx Now, taking Fourier transforms on both sides, it follows that „ « Δt “ iξ – iξ ” cn d n+1 (ξ) = 1 – U U (ξ) . a e –e 2Δx ` ´ Δt a e iξ – e– iξ is found to be larger than modulus If the symbol Θ(Δt, ξ) := 1 – 2Δx one, this means that any small perturbation in Un will be ampliﬁed in the step computing Un+1 . In our case, since e± iξ = cos ξ ± i sin ξ, we get Θ(Δt, ξ) = 1 –

Δt a i sin ξ . Δx

Taking the modulus, this amounts to „

|Θ(Δt, ξ)| = 1 + 2

« Δt 2 a sin2 ξ v 1 , Δx

regardless of how small the selected time step is. This symbol is usually called the ampliﬁcation factor of the difference scheme. The present scheme is unconditionally unstable, since the modulus of the ampliﬁcation factor is larger than one except for ξ ∈ {0, π, 2π}. Assuming a > 0, the upwind difference scheme Un+1 = Uni – i

´ Δt ` n a Ui – Uni–1 Δx

results in the sequence formulation „ « Δt a(I – S– ) Un Un+1 = I – Δx which gives the discrete Fourier transform „ « Δt d n+1 (ξ) = 1 – cn (ξ) . U a(1 – e– iξ ) U Δx

7.5 The von Neumann Stability Criterion

Let us deﬁne K :=

Δt Δx a

and compute the ampliﬁcation factor

˛ ˛ ˛ ˛ Θ(Δt, ξ) = ˛1 – K(1 – e– iξ )˛ u |1 – K| + K |e– iξ | | {z } =1 CFL condition

u

1–K+K= 1.

Hence, our upwind scheme is stable under the usual CFL condition. We remark in passing that the same analysis holds in multiple spatial dimensions. Here, the discrete Fourier transform is deﬁned as X T cn (Ξ ) = U Unk e ik Ξ k∈Zd

where d is the space dimension and Ξ = (ξ1 , . . . , ξd )T . In the case of systems of equations with n components in d spatial dimensions, ampliﬁcation is given by d n+1 cn (Ξ ) U (Ξ ) = Θ(Δt, Ξ )U

where Θ is an n ~ n matrix, the ampliﬁcation matrix. For reasons of generality, we stay with this case (systems of equations) in the following. Recursion reveals the relation c0 (Ξ ) cn (Ξ ) = Θn (Δt, Ξ )U U

so that stability is associated with the properties of the iterated ampliﬁcation matrix. In general, we can state the necessary and sufﬁcient stability criterion, that for some positive Δt∗ the matrices (or, in the case of scalar equations, the ampliﬁcation factor) Θn (Δt, Ξ ) should be uniformly bounded for 0 < Δt < Δt∗ , 0 u nΔt u T and all Ξ ∈ Zd . If ρΘ (Δt, Ξ ) denotes the spectral radius of Θ (i.e., the maximum of the moduli of the eigenvalues), then it follows from linear algebra that ρnΘ (Δt, Ξ ) u Θn (Δt, Ξ ) u Θ(Δt, Ξ )n . A necessary condition for stability is therefore the existence of a constant C such that ρnΘ (Δt, Ξ ) u C for 0 < Δt < Δt∗ , 0 u nΔt u T and all Ξ ∈ Zd . Without loss of generality, we assume that C v 1 and therefore 1

ρΘ (Δt, Ξ ) u C n ,

0unu

T . Δt

143

144

7 Types of Discretization Principles

Thus, in particular, we get Δt

ρΘ (Δt, Ξ ) u C T . Now the expression CΔt/T is bounded for 0 < Δt < Δt∗ by a linear function 1 + const Δt. Thus, a necessary condition for stability is the celebrated von Neumann condition, |λi | u 1 + O(Δt) , for 0 < Δt < Δt∗ , all Ξ ∈ Zd and i = 1, . . . , n, where λi denotes the i-th eigenvalue of Θ(Δt, Ξ ). It should be noted that there are several other stability conditions, like the famous Kreiss matrix theorem, or the much less known criterion of Buchanan. 127)

7.6 The Modiﬁed Equation

Finally, we brieﬂy describe a tool that can often help in the analysis of ﬁnite difference procedures. For every ﬁnite difference approximation of order O(Δt p , Δx q ) of a given differential equation, there is another differential equation to which the difference scheme is a better approximation. In order to demonstrate this remark, let us start again from the simple upwind scheme Un+1 = Uni – i

´ Δt ` n a Ui – Uni–1 Δx

(7.27)

for the transport equation ∂t u + a∂x u = 0 ,

a>0.

(7.28)

Inserting the Taylor series expansions for a sufﬁciently smooth function u, i.e., = u + Δt∂t u + un+1 i uni–1 = u – Δx∂x u +

Δt2 2 Δt3 3 ∂t u + ∂ u + O(Δt4 ) 2 6 t Δx2 2 Δx3 3 ∂x u – ∂ u + O(Δx4 ) , 2 6 x

127) For a thorough treatment, see R.D. Richtmyer, K.W. Morton: Difference methods for initial-value problems, 2nd ed. New York: Interscience 1967

7.6 The Modiﬁed Equation

into the upwind scheme results in ∂t u +

Δt 2 Δt2 3 Δx Δx2 3 ∂t u + ∂t u + O(Δt3 ) = – a∂x u + a ∂x2 u – a ∂ u 2 6 2 6 x + O(Δx3 ) ,

or, after rearranging terms, ∂t u + a∂x u = –

Δt 2 aΔx 2 Δt2 3 aΔx2 3 ∂t u + ∂x u – ∂t u – ∂ u 2 2 6 6 x

(7.29)

+ O(Δt3 , Δx3 ) . We now want to replace the time derivatives on the right hand side by space derivatives. In order to do so, we are not allowed to use the differential equation (7.28), since our u does not necessarily satisfy this equation, but Eq. (7.29)! Therefore, we must accomplish our task using (7.29) alone. We take the time derivative of (7.29), i.e., Δt 3 aΔx Δt2 4 ∂t u + ∂t ∂x2 u – ∂ u 2 2 6 t aΔx2 ∂t ∂x3 u + O(Δt3 , Δx3 ) , + 6

∂t2 u + a∂t ∂x u = –

and then the space derivative of (7.29), which we multiply by a, Δt 2 a2 Δx 3 aΔt2 3 ∂t ∂x u + ∂x u – ∂ ∂x u 2 2 6 t a2 Δx2 4 ∂x u + O(Δt3 , Δx3 ) . – 6

a∂t ∂x u + a2 ∂x2 u = – a

Subtracting the last equation from the ﬁrst one gives „ « „ « a 1 a a2 ∂t2 u = a2 ∂x2 u + Δt – ∂t3 u + ∂t2 ∂x u + Δx ∂t ∂x2 u – ∂x3 u 2 2 2 2 + O(Δt3 , Δx3 ) . Some other, similar, manipulations of (7.29) lead to ∂t3 u = –a3 ∂x3 u + O(Δt, Δx) , ∂x ∂t2 u = a2 ∂x3 u + O(Δt, Δx) , ∂t ∂x2 u = –a∂x3 u + O(Δt, Δx) . Substituting these expressions into (7.29) yields the modiﬁed equation „ « „ « aΔx Δt aΔx2 Δt2 Δt 2 1– ∂x u – 2 2 –3 + 1 ∂x3 u ∂t u + a∂x u = 2 Δx 6 Δx Δx + O(Δx3 , Δx2 Δt, ΔxΔt2 , Δt3 ) .

145

146

7 Types of Discretization Principles

What can be seen from this equation? First of all, we see that the truncation error of our difference scheme applied to transport equation (7.28) is O(Δt, Δx), since the ﬁrst term on the right side of the modiﬁed equation is a a Δx∂x2 u – Δt∂x2 u 2 2 which goes to zero linearly as time and space increase. Moreover, our upwind scheme is a good approximation to the parabolic differential equation ∂t u + a∂x u = ε(a, Δt)∂x2 u with a positive (due to the CFL condition) diffusion coefﬁcient. Since such equations have smooth solutions, we expect our upwind scheme to be well behaved. Many further properties of the difference scheme can be deduced from the modiﬁed equation, e.g., numerical stability.

7.7 Difference Schemes in Conservation Form

It seems obvious that numerical procedures can only be expected to lead to good approximations of the solution to the original problem if the scheme imitates as many of the properties known in advance as possible. One of these reproductions, already mentioned in Chapter 6, is the use of socalled methods of conservation form in order to treat conservation law equations numerically. This means that the system of conservation laws (2.1), i.e., ∂t V + ∂x f (V ) = ∂t V + J f (V ) ∂x V = 0 is approximated by an explicit (2k + 1)-step ﬁnite difference scheme ” ” 1 “ n 1 “ n+1 Vj – Vjn + gj+ 1 – gnj– 1 = 0 2 2 Δt Δx with a consistent numerical ﬂux “ ” n n n n , Vj–k , . . . , Vjn , Vj+1 , . . . , Vj+k gnj+ 1 := g Vj–k+1 2

(7.30)

(7.31)

(7.32)

and not (for example) by a method like ” ” 1 “ n+1 1 “ n Vj – Vnj + J f (Vnj ) Vj+1 – Vnj–1 = 0 . Δt 2 Δx An example of a method of type (7.31) is the three-point Lax–Friedrichs scheme 128) Vjn+1 –

” 1 Δt n o 1“ n n n n Vj+1 + Vj–1 + f (Vj+1 ) – f (Vj–1 ) =0 2 2 Δx

128) K.O. Friedrichs, P.D. Lax: Proc. Nat. Acad. Sci. USA 68 (1971) 1686–1688

(7.33)

7.7 Difference Schemes in Conservation Form

with gj+n 1 = – 2

” 1n o 1 Δx “ n n Vj+1 – Vjn + f (Vj+1 ) + f (Vjn ) . 2 Δt 2

If the numerical vectors Vnj in (7.33) are replaced by an exact smooth solution V(xj , tn ) from (7.30), the left side of (7.33) differs from zero, but its Taylor expansion then leads to a truncation error of order O(Δt) in the sense of the Lax–Richtmyer Δt theory, provided that λ = Δx = const is prescribed. Hence, (7.33) is a method of conservation form of order one. It can be shown that the method is monotone according to the meaning given in Section 6.3, and, as a matter of fact, all monotone methods are of order one, as can be established using the “modiﬁed equation” in Section 7.6. However, there are of course also methods of higher order. One of them is the three-point Lax–Wendroff scheme. 129) For scalar one-dimensional conservation laws of type (2.1), its numerical ﬂux is given by j “ ” “ ” 1 “ ”2 “ ”ﬀ 1 n n f vj+1 + f vjn – μnj+ 1 vj+1 – vjn (7.34) gj+n 1 = 2 2 2 λ with μnj+ 1 = λ 2

n ) – f (vjn ) f (vj+1 n vj+1 – vjn

, λ=

Δt . Δx

By Taylor expansion, it can be veriﬁed that this is a method of order two. Therefore, the scheme seems to be a useful one, at least as long as the smoothness assumptions necessary for the Taylor expansion are guaranteed in the neighborhood of (xj , tn ). However, if the exact weak solution shows a shock at a certain position, the numerical solution begins to oscillate close to this position, although the scheme respects the conservational character of the original problem. What is not respected by the method is the TVD property of the solutions to the original problem. Numerical solutions produced via TVD methods do not begin to oscillate, because this would lead to an increase in the total variation. Several attempts were made in the literature to overcome the problem of low monotone method order and retain the advantages of higher order methods, e.g., the use of greater step sizes without any loss of accuracy in areas where the exact solution is smooth. One of these ideas, e.g., for scalar problems, involves constructing so-called ﬂux limiter methods. Let gH be the numerical ﬂux of a higher order method, e.g., of the Lax–Wendroff scheme, and let gL be the numerical ﬂux of a low-order procedure, e.g., a TVD scheme. 129) P.D. Lax, B. Wendroff: Comm. Pure Appl. Math. 13 (1960) 217–237

147

148

7 Types of Discretization Principles n n If gH (vj+k–1 , . . . vj+k ) is abbreviated by gH (v n ; j) and analogously gL (v n ; j), let us create a new method via

g(v n ; j) := gL (v n ; j ) + ϕ(v n ; j) {gH (v n ; j ) – gL (v n ; j )} .

(7.35)

Here, ϕ – called the limiter – must be chosen in such a way that ϕ W 1 in areas where the exact entropy solution is expected to be smooth, and ϕ W 0 close to shocks. In other words, ϕ has to be a sensor for sudden strong increases or decreases in the unknown exact solution that are expected to occur at positions where the numerical solution increases or decreases strongly with respect to space, e.g., ! n vjn – vj–1 ϕ=ϕ (7.36) n vj+1 – vjn where ϕ is a sufﬁciently smooth and bounded function with ϕ(1) = 1 and ϕ(0) = 0. Of course, difﬁculties can still cause trouble, e.g., if an extremum of the solution lies between xj and xj+1 . Other ideas have also been devised to surmount the problem of nonphysical oscillations when higher-order methods are being considered. One example used for higher dimensions are the so-called ENO schemes (essentially non-oscillatory schemes). 130) This idea involves piecewise polynomial reconstruction from the piecewise constant values of the approximate solution for the last time step, where the stencil for each cell is chosen adaptively in such a way that intensive oscillations do not occur near shocks.

7.8 The Finite Volume Method on Unstructured Grids

Let us ﬁnally brieﬂy put the idea of ﬁnite volume schemes on unstructured grids, introduced in Section 7.1, into concrete form. For convenience, we restrict ourselves to the scalar two-dimensional conservation law (cf. (1.10)) ∂t u + ∂x f1 (u) + ∂y f2 (u) = 0 in R2 ~ R+ , u(x, y; 0) = u0 (x, y) in R2 .

(7.37)

ˆ ⊂ R2 consists of a set of polygons σi (i ∈ I ⊂ N), A so-called unstructured grid of Ω e.g., triangles, each of which has the same number m of edges and vertices, so that [ ˆ = Ω σi , i∈I

and two polygons are not assumed to intersect, or to only intersect along a common edge or at a common vertex. These polygons are often called cells. The angles at the 130) See A. Harten: Multi-dimensional ENO schemes for general geometries (ICASE Report 91-76). Hampton: Langley Research Center 1991

7.8 The Finite Volume Method on Unstructured Grids

vertices must be bounded uniformly away from zero, and we assume h = O(Δt), where h represents a measure of the maximal diameter of the cells. Unstructured grids can be more easily adapted to complicated geometries than structured grids (e.g., to the gas ﬂow around ﬂying objects), and reﬁnements of the net in regions where the exact solution is expected to vary more than in neighboring areas are easily realized. The development of grid generation techniques is therefore a very important task in this context. Let Ii ⊂ I be the index set consisting of the numbers of the particular cells that are neighbors of σi with a joint edge. With un (x, y) := u(x, y; n Δt) (n ∈ N0 ) in the case of a given time step Δt, and if an exact solution u is assumed to be smooth for a short time, Eq. (7.37) leads to un+1 (x, y) – un (x, y) + ∂x f1 (un (x, y)) + ∂y f2 (un (x, y)) = O(Δt) . Δt

(7.38)

Taking the divergence theorem into account, integration of this equation over the cell σi with edges ∂σi,k (k = 1, 2, . . . , m) results in Z D E XZ ˘ n+1 ¯ 1 u (x, y) – un (x, y) dσi + f (un ), ni,k ds = O(Δt) (7.39) Δt σi ∂σi,k k∈Ii

`

´T

where f (u) = f1 (u), f2 (u) is assumed to be sufﬁciently smooth, and where ni,k denotes the outer normal unit vector on ∂σi,k . 131)132) If we replace the functions un (x, y), un+1 (x, y) along the cell σi by constant approximate values vin , vin+1 (to be calculated), if the term O(Δt) is neglected, and if |σi | denotes the area of σi and si,k the length of the edge ∂σi,k , (7.39) leads to vin+1 = vin –

Δt X D ` n ´ i,k E f v˜i,k , n si,k |σi |

(7.40)

k∈Ii

n (n = 0, 1, 2, . . .), where the values v˜i,k represent approximations for u along the edges ∂σi,k , respectively, and where vi0 is chosen to be the mean value of the initial function on σi : Z 1 vi0 = u0 (x, y) dσi , ∀ i ∈ I . |σi | σi

n ), ni,k si,k approximately Obviously, from a physical point of view, the term f (˜ vi,k describes the ﬂux through the edge ∂σi,k . n The question of how the values v˜i,k should be chosen then arises. Because the conservation principle should also be reﬂected by the approximate values, we aim for D ` ´ E D ` ´ E n n f v˜i,k , ni,k si,k W – f v˜k,i , nk,i sk,i (sk,i = si,k ) . (7.41)

131) See (7.4) 132) Here, we assume that each of the cells has neighbors along each of its edges. If this is

not the case, namely parts of the boundary ˆ are along the edges, appropriate of Ω boundary conditions must be respected.

149

150

7 Types of Discretization Principles n In order to create a scheme that is easy to handle, we are going to replace v˜i,k n n n and v˜k,i by a certain mean value of vi and vk and replace the physical ﬂux with a numerical ﬂux gi,k (vin , vkn ) that fulﬁlls relation (7.41) exactly, i.e.,

gi,k (vin , vkn ) = –gk,i (vkn , vin ) .

(7.42)

Moreover, gi,k (w, w) = f (w), ni,k si,k

(7.43)

should be fulﬁlled, making the ﬁnite volume method that is now constructed, namely vin+1 = vin –

Δt X gi,k (vin , vkn ) , |σi |

(7.44)

k∈Ii

consistent with (7.39). Finally, we now forget about the assumption regarding the smoothness of the exact solution, because the integration process also allows us to deal with weak solutions. Convergence proofs for FVMs in higher dimensions and on unstructured grids, particularly mathematical proofs for convergence to the entropy solutions, are often not currently available. However, it is possible to formulate the methods in such a way that they lead to well-known methods that work well if analogously formulated for one-dimensional problems. One method that can be shown to behave very well in the 1D situation is the Engquist–Osher ﬂux splitting scheme 133) with a numerical ﬂux described by Eq. (6.24). If the numerical ﬂux gi,k in (7.43) is chosen in accord with (6.24) to be h i v) gi,k (v, vˆ) = ˆfi,k– (v) + ˆfi,k+ (ˆ

(7.45)

with ˆfi,k (w) := f (w, ni,k si,k and with ˆf + (v) = i,k ˆf – (ˆ i,k v )

Z Z

v

n o max ˆf i,k (w), 0 dw + ˆfi,k (0)

vˆ

n o min ˆf i,k (w), 0 dw ,

0

= 0

(7.46)

it corresponds to the Engquist–Osher scheme in 1D, and it fulﬁlls the relations (7.42) and (7.43). 133) See Section 6.3

7.9 Continuous Convergence of Relations

7.9 Continuous Convergence of Relations

Most of the convergence concepts for discretization procedures mentioned within this book can be considered realizations of an easy general concept called continuous convergence of relations. The convergence results for the Lax–Richtmyer theory (cf. Theorem 7.1) and the convergence proofs in connection with Rinow’s theorem (cf. Theorem 6.3) or with the general considerations of Section 6.3 can be regarded as concretizations of this concept, which is based on the ideas of P. du Bois-Reymond, R. Courant and W. Rinow already mentioned in footnotes 103–105. Roughly speaking, the continuous convergence of objects Pn acting on elements un (n = 1, 2, . . .) to an object P acting on an element u implies un → u ∧ Pn un → v ⇒ Pu = v .

(7.47)

We write c

Pn → P . 134) The general concept reads as follows: 135) Assume that there is a metric space V, an index set J and a relation R such that, for ordered pairs (u, Φ) ∈ V ~ J , the question of whether (u, Φ) belongs to R ⊂ V ~ J or not can be answered. If the relation is fulﬁlled, i.e., if (u, Φ) ∈ R, we write uRΦ . Moreover, asume that there are subsets Vn ⊂ V (n = 1, 2, . . .) and relations Rn concerning ordered pairs (un , Φ) ∈ Vn ~ J (n = 1, 2, . . .) such that the question of whether or not un is related to Φ arises. If the answer is “yes,” i.e., if (un , Φ) ∈ Rn , we write un Rn Φ . Deﬁnition We call the sequence {Rn } of relations continuously convergent to the relation R with respect to the triple (V, {Vn } , J ) if the following implication holds: ∀ Φ ∈ J : {un |un ∈ Vn (n = 1, 2, . . .) ; un Rn Φ} → u ⇒ uRΦ .

(7.48)

We then write c

Rn → R . 134) This is a little more general than the deﬁnition of continuously convergent sequences of operators mentioned in Section 6.3.

135) R. Ansorge: ZAMM 86 (2006) 656–664 136) Numerical equations which do not have a solution do not make sense.

151

152

7 Types of Discretization Principles

Considering our focus on weakly formulated problems that do not necessarily have unique solutions, it could be interesting to see how this general concept ﬁts with this situation, which is the situation discussed in Sections 6.2 and 6.3. Assume that there is a given problem Au = v

(7.49)

where A maps a metric space V into a metric space W with a given right hand side v ∈ W. We ask for solutions u ∈ V of problem (7.49). Let S = {u ∈ V|u solves (7.49)} . Provided that S contains more than one element, we ask for a unique entropy solution uE ∈ S. For this reason, we take the ordered pairs {(u, Φ)|u ∈ S, Φ ∈ V} into account where J is a suitable index set, and let R be a relation between elements of S and J so that there is at most one element in S, denoted by uE (if it exists), which fulﬁlls the entropy condition uE RΦ ,

∀Φ ∈ J .

(7.50)

Hence, the complete problem to be solved consists of ﬁnding a solution uE to problem (7.49) which additionally fulﬁlls the relation (7.50). Let Vn ⊂ V (n = 1, 2, . . .) be a sequence of subspaces and let {v n |v n ∈ Vn , (n = 1, 2, . . .)} ⊂ V be a given sequence with lim v n = v .

(7.51)

n→∞

The numerical procedure for solving (7.49) approximately then consists of constructing operators An : Vn → W, (n = 1, 2, . . .) such that each of the problems A n un = v n ,

(n = 1, 2, . . .)

(7.52)

has at least one solution, 136) i.e., Sn := {un ∈ Vn |un solves (7.52)} =/ ∅ for each ﬁxed n ∈ N . Here, it can happen that Sn contains more than one element, but let Sn → S in the sense of the set convergence described by the last deﬁnition in Section 6.2. Then the following theorem can be stated: Theorem 7.2 For each ﬁxed n ∈ N , let Rn be a relation between elements of Sn and of J , and assume that – There is at least one element unE ∈ Sn with unE Rn Φ, ∀ Φ ∈ J , (n = 1, 2, . . .) c – Rn → R .

7.9 Continuous Convergence of Relations

Then there is a unique entropy solution uE to the problem {(7.49), (7.50)}, and each full sequence {unE |unE ∈ Sn ,

(n = 1, 2, . . .)}

(7.53)

converges to uE . Proof: Because of the assumptions, there is at least one sequence of type (7.53), and each of these sequences is of the type {un |un ∈ Sn , (n = 1, 2, . . .)}. Take one of the convergent subsequences of a particular type-(7.53) sequence and denote its limit c as uE . Sn → S implies uE ∈ S, and Rn → R yields uE RΦ ,

∀Φ ∈ J .

However, because there is at most one element uE ∈ S which fulﬁlls (7.49), uE is unique and depends on neither the particular type-(7.53) sequence nor the special convergent subsequence of this sequence. Also, because of the uniqueness of uE , not only subsequences but also the full sequence converge to this limit. It is easy to identify the abstract relations used in the terminology of this section with the different concrete mathematical objects of previous convergence results. However, the concept of continuous convergence of relations also applies to other ﬁelds of applied mathematics, e.g., to approximation theory.

153

155

8 A Closer Look at Discrete Models 8.1 The Viscosity Form

We have already brieﬂy discussed Riemann problems and their meaning for numerical purposes in Chapter 4. Now we shall identify the Riemann problem as one of the central issues of discrete models of hyperbolic conservation laws. We start with a discussion of the scalar case 137) and then turn towards the Riemann problem for the Euler equations. The building block for any ﬁnite difference model of scalar conservation laws ∂t u + ∂x f(u) = 0 with initial conditions u(x, 0) = u0 (x) is a robust ﬁrst-order numerical ﬂux function n gnj+ 1 = g(vj+1 , vjn ) to be used in the conservation form 2

vjn+1 = vjn –

” Δt “ n gj+ 1 – gnj– 1 , 2 2 Δx

(8.1)

cf. (6.14). In contrast to (6.15), we derive a theory for three-point numerical ﬂuxes, since three points are sufﬁcient for ﬁrst-order accuracy. We have already encountered the ﬁrst-order monotone Lax–Friedrichs scheme (7.33) deﬁned through the numerical ﬂux ” 1n o Δx “ n n gnj+ 1 := vj+1 – vjn + f(vj+1 ) + f(vjn ) . (8.2) 2 2Δt 2 The form in which this numerical ﬂux has been written is not very instructive. We will therefore exploit the idea of considering a discrete model to be a discretization of the perturbed equation ∂t u + ∂x f(u) = ∂x (Q(u)∂x u) .

(8.3)

The right hand side perturbation models a stabilizing diffusion term in which the function u → Q(u) is called the viscosity coefﬁcient. It is one of the most important 137) See E. Tadmor: Math. Comp. 43 (1984) 353–368; E. Tadmor: Math. Comp. 43 (1984) 369–381

156

8 A Closer Look at Discrete Models

and striking consequences of the theory we shall study that some sort of viscosity is necessary in order to build stable discrete models. With (8.3) in mind, a discrete numerical ﬂux should always have the viscosity form ” Δx 1“ n n gnj+ 1 = f(vj ) + f(vj+1 ) + (8.4) Qn 1 (v n – v n ) 2 2 2Δt j+ 2 j+1 j n ). As a consequence, the differwith a numerical viscosity coefﬁcient Qnj+ 1 = Q(vjn , vj+1 2 ence ” Δx n o 1“ n n n f(vj+1 ) – f(vjn ) + Qnj+ 1 · (vj+1 – vjn ) – Qnj– 1 · (vjn – vj–1 ) gnj+ 1 – gnj– 1 = 2 2 2 2 2 2Δt clearly shows the reason for using (8.3) as a model equation. The case Qn 1 = 0, j+ 2

i.e., no viscosity, results in the central difference of the ﬂuxes, which is – after linearization – shown to be unconditionally unstable in von Neumann’s stability theory (cf. Section 7.5). One could now argue that any positive function Q (however small) could lead to stable schemes, but – as we shall see – this is by no means true!

8.2 The Incremental Form

In Section 6.3 we introduced TVD and monotone methods of type (8.1). Method (8.1) is said to be TVD if ˛ X˛ ˛ X ˛˛ ˛ ˛ n ˛ n+1 – vjn+1 ˛ u ∀n ∈ N : ˛vj+1 ˛vj+1 – vjn ˛ . j

j

We call methods of type (8.1) monotonicity preserving if v.n monotone ⇒ v.n+1 monotone

(8.5)

holds. We can now easily prove that (8.1) monotone ⇒ (8.1) TVD ⇒ (8.1) monotonicity preserving

(8.6)

holds. One can easily show that solutions to monotone schemes form l1 -contractive semigroups, i.e., that n+1 n v.+1 – v.n+1 l1 (R) u v.+1 – v.n l1 (R) P is valid, where v.n l1 (R) := j |vjn |. Hence, the ﬁrst implication follows. To prove the second, we consider the grid function

vjn vjn vjn

=

const =: vLn ;

j< k–1

monotonically increasing ; =

const =:

vRn

; |vRn

k–1u ju k+1

j >k+1.

In that case, the total variation is – vLn |. Assume that v.n+1 does not monotonn+1 n+1 ically increase, so that we have a maximum and a minimum – say vM and vm ,

8.2 The Incremental Form

respectively. It would follow that X n+1 n+1 n+1 TV(v.n+1 ) := |vj+1 – vjn+1 | v |vRn – vLn | + |vM – vm | > TV(v.n ) , j

which contradicts the TVD property. This concludes the proof. The incremental form of a scheme (8.1) can be achieved if there are incremental factors C± , so that (8.1) can be written as j+ 1 2

vjn+1

n n = vjn + Cj++ 1 · (vj+1 – vjn ) – Cj–– 1 · (vjn – vj–1 ). 2

(8.7)

2

It is an easy exercise to show that every scheme (8.1) can be put into incremental form, and that the incremental factors are given by Cj++ 1 = 2

Cj–– 1 2

n n Δt f(vj ) – gj+ 12 n Δx vj+1 – vjn

n n Δt f(vj ) – gj– 12 = . n Δx vjn – vj–1

Theorem 8.1 Monotonicity-preserving three-point schemes satisfying Cj++ 1 v 0 ;

Cj+– 1 v 0 ;

2

1 – Cj++ 1 – Cj+– 1 v 0

2

2

(8.8)

2

are TVD schemes. Proof: Applying difference operators Δ to (8.7) yields “ ” n n Δvjn = Cj++ 1 Δvj+1 + 1 – Cj++ 1 – Cj+– 1 Δvjn + Cj–– 1 Δvj–1 . 2

2

2

(8.9)

2

n n In the case of Δvj+1 = Δvj–1 = 0, it follows that

“ ” Δvjn+1 = 1 – Cj++ 1 – Cj+– 1 Δvjn . 2

2

Preserving monotonicity means sign Δvjn+1 = sign Δvjn , and hence 1 – Cj++ 1 – Cj+– 1 v 0 . 2

2

The two remaining inequalities follow analogously. Summing (8.9) yields X X X n n TV(v.n+1 ) u |Cj++ 3 ||Δvj+1 |+ |1 – Cj++ 1 – Cj+– 1 ||Δvjn | + |Cj–– 1 ||Δvj–1 | j

2

j

2

2

and an index shift in the ﬁrst sum and third sum reveals TV(v.n+1 ) u TV(v.n ).

j

2

157

158

8 A Closer Look at Discrete Models

8.3 Relations

With an eye on (8.1) with (8.4), as well as on (8.7), we can identify the numerical viscosity coefﬁcient as Qnj+ 1 = Cj++ 1 + Cj+– 1 = 2

2

2

n n n Δt f(vj ) + f(vj+1 ) – 2gj+ 12 . n Δx vj+1 – vjn

(8.10)

Conversely, the incremental factors can be represented as C± j+ 12

1 = 2

Qnj+ 1 2

n n Δt f(vj+1 ) – f(vj ) ∓ n Δx vj+1 – vjn

!

.

(8.11)

By exploiting relations (8.10) and (8.11), we can already characterize difference schemes (8.1) by their numerical viscosity coefﬁcients. Theorem 8.2 Difference schemes (8.1) with numerical ﬂux (8.4) are TVD schemes if ˛ ˛ n n Δt ˛˛ f(vj+1 ) – f(vj ) ˛˛ ˛ ˛ u Qnj+ 1 u 1 n 2 Δx ˛ vj+1 – vjn ˛

(8.12)

holds true. Proof: If (8.12) holds, then the incremental factors satisfy the assumptions of Theorem 8.1. Theorem 8.3 If Δt Δx

˛ n ˛ ˛ f(v ) – f(v n ) ˛ 1 j ˛ ˛ j+1 ˛ ˛ u Qnj+ 1 u n n ˛ vj+1 ˛ 2 – vj 2

(8.13)

holds, then the maximum principle inf vkn u vkn+1 u sup vkn

∀k ∈ Z :

k

(8.14)

k

follows. Proof: From (8.7), one derives n n vjn+1 = Cj++ 1 vj+1 + (1 – Cj++ 1 – Cj–– 1 )vjn + Cj+– 1 vj–1 2

2

2

2

8.4 Godunov Is Just Good Enough n n which is a convex combination of vj+1 , vjn , vj–1 . Now (8.13) follows because of

0u

C± j+ 12

1 = 2

n

Qnj+ 1 2

n

Δt f(vj+1 ) – f(vj ) ∓ n Δx vj+1 – vjn

!

u

1 1 · 2Qnj+ 1 u . 2 2 2

n –vjn = 0. As can be seen from (8.12), We have not yet discussed the case when vj+1 in this case the numerical viscosity coefﬁcient has to satisfy the inequality

Δt n | f (vj )| u Qnj+ 1 u 1 , 2 Δx or, from (8.13), Δt n 1 | f (vj )| u Qnj+ 1 u 2 Δx 2 in order to save the maximum principle. This, of course, is simply a type of CFL condition (cf. Section 7.3). Theorems 8.2 and 8.3 clearly deﬁne the range of proper numerical viscosity coefﬁcients, and one could request the two difference schemes that satisfy Qnj+ 1 = 1 and 2 ˛ ˛ n n Δt ˛˛ f(vj+1 ) – f(vj ) ˛˛ n Q j+ 1 = ˛ ˛, respectively. n 2 Δx ˛ vj+1 – vjn ˛ These schemes do actually exist, and we have already encountered the one with the highest numerical viscosity possible: setting Qnj+ 1 = 1 in (8.4) leads to 2

” Δx 1“ n n = f(vj ) + f(vj+1 ) + (v n – v n ) , 2 2Δt j+1 j

gnj+ 1 2

and this is simply the Lax–Friedrichs ﬂux (8.2), i.e., Theorem 8.4 The Lax–Friedrichs scheme is characterized by Qnj+ 1 =: QLF =1. j+ 1 2

2

(8.15)

8.4 Godunov Is Just Good Enough

Surprisingly, the scheme with the least possible numerical viscosity coefﬁcient has also been known for quite a long time. In 1952 Courant, Isaacson and Rees published a paper 138) in which they derived the classical Courant–Isaacson–Rees ﬁnite difference scheme. Unfortunately, their 138) R. Courant, E. Isaacson, M. Rees: Comm. Pure Appl. Math. 5 (1952) 243–244; see also E.M. Murman: AIAA J. 12 (1974) 626–633 and footnote 87

159

160

8 A Closer Look at Discrete Models

scheme was not formulated in conservation form, but this was remedied by Murman in 1974. The Murman–Courant–Isaacson–Rees (MCIR) scheme is given by “ ” “ ”” Δt “ n n vjn+1 = vjn – sj+ 1 f(vj+1 ) – f(vjn ) + (1 – sj– 1 ) f(vjn ) – f(vj–1 ) 2 2 Δx where !! n f(vj+1 ) – f(vjn ) 1 sj+ 1 := 1 – sign . n 2 2 vj+1 – vjn It is easily seen that the numerical ﬂux of this difference scheme is given by ˛ ˛ ” 1 ˛ f(v n ) – f(v n ) ˛ 1“ n j ˛ ˛ j+1 n n n f(vj ) + f(vj+1 ) – ˛ – vjn ) , gj+ 1 = ˛ · (vj+1 n 2 2 2 ˛ vj+1 – vjn ˛ and so the numerical viscosity coefﬁcient is given by ˛ n ˛ n Δt ˛˛ f(vj+1 ) – f(vj ) ˛˛ MCIR Q j+ 1 := ˛ ˛ . n 2 Δx ˛ vj+1 – vjn ˛

(8.16)

According to Theorem 8.2, this is the least possible numerical viscosity coefﬁcient for a TVD scheme. There is, however, a serious problem with the MCIR scheme. In the case of Burgers’ equation „ « 1 2 ∂t u + ∂x u =0 2 with the initial condition ( –1 , u(x, 0) = u0 (x) = 1, i.e.,

(

vj0

=

–1 , 1,

x<0 , xv0

j<0 , jv0

the scheme gives a stable, entropy-violating stationary shock. The reason for this behavior is easily found: the numerical viscosity coefﬁcient vanishes in the rarefaction shock. We therefore have to search for a scheme with a nonvanishing viscosity coefﬁcient resembling (8.16). This is the famous Godunov scheme. According to (4.1), we now think of initial values as being piecewise constant on an initial grid, i.e., Z xj +xj+1 2 1 0 vj := u0 (x) dx , j ∈ Z . Δx xj–12+xj Hence, the initial values give rise to local Riemann problems at points . . . , xj– 1 , 2 xj+ 1 , . . . Godunov’s brillant idea was to solve the local Riemann problems exactly 1 and use the solution as new values at the next time step.

8.4 Godunov Is Just Good Enough

t Δt

j-1

j

j+1

x

Fig. 8.1 Local Riemann problems.

Deﬁnition Godunov’s method is given by the recipe vjn+1 =

1 – (u 1 + uj++ 1 ) 2 2 j– 2

(8.17)

where Δx

uj–– 1 2

2 := Δx

Z2

n uR (z, tn+1 ; vj–1 , vjn ) dz

(8.18)

0

and uj++ 1 := 2

2 Δx

Z0

n uR (z, tn+1 ; vjn , vj+1 ) dz ,

(8.19)

– Δx 2

and uR (z, tn+1 ; uL , uR ) denotes the exact solution of the Riemann problem at time tn+1 with initial left and Δx right data uL and uR , respectively. Here, z is a local coordinate z ∈ [ –Δx 2 , 2 ] around xj– 1 and xj+ 1 , respectively. 2

2

In order to avoid interactions between the waves from neighboring Riemann solutions, we have to admit a slightly stronger CFL condition: ˛ ˛ 1 Δt ˛ ˛ max ˛f (vjn )˛ u . (8.20) Δx j 2 Integrating our differential equation over a “cell” [xj– 1 , xj+ 1 ] gives 2

xj+ 1

xj+ 1

xj– 1 tn

xj– 1 tn

Z 2 Ztn+1 Z 2 Ztn+1 R ∂t u dt dx + ∂x f(uR ) dt dx = 0 2

2

2

161

162

8 A Closer Look at Discrete Models

u -j-1/2

vj-1

u +j+1/2

vj

xj-1

vj+1

vj

x j-1/2

xj

xj+1/2

xj+1

Fig. 8.2 Local Riemann problems.

and hence Δx

Z2

Z0

u

R

n (z, tn+1 ; vj–1 , vjn )

(8.21)

– Δx 2

0 Δx

Z2

–

n uR (z, tn ; vj–1 , vjn ) dz –

Z0

n uR (z, tn ; vjn , vj+1 ) dz

– Δx 2

0

+

n uR (z, tn+1 ; vjn , vj+1 ) dz

dz +

Ztn+1n o n n f(uR (0, t; vjn , vj+1 )) – f(uR (0, t; vj–1 , vjn ) dt = 0 . tn

According to (8.17)–(8.19), the sum of the two leading integrals is Δx · vjn+1 , while the sum of the following two is Δx · vjn . The Riemann solution is constant on rays, n so uR (0, . . .) is a constant value. Let us denote this value of uR (0, t; vj–1 , vjn ) by wj– 1 , 2 R n n and the value u (0, t; vj , vj+1 ) by wj+ 1 . Then the remaining integral in (8.21) evalu2 ates to “ ” Δt f(wj+ 1 ) – f(wj– 1 . 2

2

We have therefore shown in passing that the Godunov difference scheme can be written as ” Δt “ G vjn+1 = vjn – gj+ 1 – gG 1 j– 2 2 Δx with the numerical ﬂux := f(wj+ 1 ) . gG j+ 1 2

2

(8.22)

n Hence, if there is no state v between vjn and vj+1 for which f (v) = 0, the term wj+ 1 2 n n will take the value vj+1 or vj according to where the wave lies. In this case, the

8.4 Godunov Is Just Good Enough

numerical Godunov ﬂux can be written as ˛ n ˛ ! ˛ f(v ) – f(v n ) ˛ 1 j ˛ n ˛ j+1 n n n gG = f(v ) + f(v ) – (v – v ) ˛ ˛ j j+1 n j+ 12 ˛ vj+1 2 – vjn ˛ j+1 j

(8.23)

and the viscosity coefﬁcient (8.16) of the MCIR scheme can be identiﬁed immediately. The Godunov scheme and the MCIR scheme are identical in this case. In the case of a wave with f (u) = 0, we can get additional information from (8.10). This gives the following formula for the full Godunov viscosity coefﬁcient: 8 n n G > > Δt f(vj ) + f(vj+1 ) – 2gj+ 12 > n > ; f (vjn ) < 0 < f (vj+1 ) > n n < Δx v – v j+1 j G Qj+ 1 = (8.24) ˛ n ˛ n 2 > > Δt ˛˛ f(vj+1 ) – f(vj ) ˛˛ > > > ; otherwise : Δx ˛˛ v n – v n ˛˛ j+1 j from (8.22). Hence, the Godunov scheme has a numerical viswhere we take gG j+ 1 2

cosity coefﬁcient that never vanishes. Let us now check the behavior of the Godunov scheme with respect to an entropy inequality. As in previous chapters – see Section 3.2 – we denote a convex entropy function by S and a corresponding entropy ﬂux 139) by F. A function u solving a conservation law is called entropy solution if the entropy inequality ∂S ∂F (u) + (u) u 0 ∂t ∂x holds for every pair (S, F) in the weak sense. As above, we integrate Δx

Z 2 Ztn+1 0

tn

Δx

∂S R (u ) dt dx + ∂t

Z 2 Ztn+1 0

tn

∂F R (u ) dt dx u 0 ∂x

and get (cf. (8.21)) Δx

Z2

n S(uR (z, tn+1 ; vj–1 , vjn )), dz u 0

“ ” Δx S(vjn ) – Δt F(vjn ) – GG 1 j– 2 2

where GG = F(wj– 1 ) j– 1 2

2

is a consistent numerical entropy ﬂux. Completely analogously, we get Z0 – Δx 2

n S(uR (z, tn+1 ; vjn , vj+1 )) dz u

“ ” Δx . S(vjn ) + Δt F(vjn ) – GG 1 j+ 2 2

139) For the relation between entropy and entropy ﬂux, see (3.13) 140) See, for example, E. Hewitt, K. Stromberg: Real and abstract analysis. Berlin: Springer 1965

163

164

8 A Closer Look at Discrete Models

Since an entropy function is always convex, we can use Jensen’s inequality, 140) which gives Δx

S(uj–– 1 ) 2

2 u Δx

Z2

n S(uR (z, tn+1 ; vj–1 , vjn )) dz 0

and S(uj++ 1 ) u 2

2 Δx

Z0 n S(uR (z, tn+1 ; vjn , vj+1 )) dz – Δx 2

Employing (8.17) ﬁnally yields ” ” 1“ Δt “ G S(uj–– 1 ) + S(uj++ 1 ) u S(vjn ) – Gj+ 1 – GG . (8.25) S(vjn+1 ) u 1 j– 2 2 2 2 2 Δx Hence, Godunovs method satisﬁes every discrete entropy inequality. We ﬁx our achievements in the following theorem Theorem 8.5 Under the CFL condition (8.20), it holds that ” n n f(vjn ) – f(vj–1 ) – QGj– 1 (vjn – vj–1 ) 2 2 “ ” n n u+j+ 1 f(vj+1 ) – f(vjn ) + QGj+ 1 (vj+1 – vjn ) 2 2 ” Δt “ n S(uj–– 1 ) u S(vjn ) – 2 F(vj ) – GG 1 j– 2 2 Δx ” Δt “ n + n S(uj+ 1 ) u S(vj ) + 2 F(vj ) – GG 1 j+ 2 2 Δx

Δt Δx Δt = vjn – Δx

uj–– 1 = vjn –

“

(8.26) (8.27) (8.28) (8.29)

Proof: The entropy inequalities (8.28) and (8.29) have just been proven. The formulae (8.26) and (8.27) can easily be veriﬁed. We have proven that the Godunov scheme is the three-point scheme with the least amount of numerical viscosity possible according to Theorem 8.3. We will see soon that the Lax–Friedrichs scheme – the scheme with maximum possible viscosity – is closely related to the Godunov scheme.

8.5 The Lax–Friedrichs Scheme

We follow Tadmor 141) and rewrite the classical Lax–Friedrichs scheme ” Δt “ ” 1“ n n n n vjn+1 = vj+1 + vj–1 – f(vj+1 ) – f(vj–1 ) 2 2Δx 141) See footnote 137

8.5 The Lax–Friedrichs Scheme

t=2Δ t

t= Δ t t=0 (j-2)Δ x

jΔ x

(j+2)Δ x

Fig. 8.3 Staggered grid.

with numerical ﬂux 1 = 2

gLF j+ 12

„

f(vjn )

+

n f(vj+1 )

Δx n – (v – v n ) Δt j+1 j

«

and numerical viscosity coefﬁcient =1 QLF j+ 1 2

so that it appears as a ﬁnite difference scheme on a staggered grid like that shown in Fig. 8.3. The initial function is now assumed to be constant between [(j – 2)Δx, jΔx], [jΔx, (j + 2)Δx], etc. At the next time step, we must discretize such that we get constant values between [(j – 1)Δx, (j + 1)Δx], etc. To avoid wave interactions, we now have to satisfy the CFL condition Δt max | f (vjn )| u 1 , Δx j which is exactly the natural CFL condition for the Lax–Friedrichs scheme. Surprisingly, the following result holds. Theorem 8.6 The following lemma is known as Tadmor’s lemma. The Godunov scheme on the staggered grid described above is exactly the Lax–Friedrichs scheme, i.e., vjn+1

1 = 2Δx

ZΔx n n uR (z, tn+1 ; vj–1 , vj+1 ) dz = –Δx

n n vj+1 + vj–1

2

–

Δt n n ) – f(vj–1 )) . (f(vj+1 2Δx

165

166

8 A Closer Look at Discrete Models

Proof: We proceed along the lines already laid out above. Integration of the conservation law gives ZΔx

ZΔx n n uR (z, tn+1 ; vj–1 , vj+1 ) dz –

–Δx

n n uR (z, tn ; vj–1 , vj+1 ) dz

–Δx

Ztn+1 Ztn+1 R n n n n , vj+1 )) dt = 0 . + f(u (Δx, t; vj–1 , vj+1 )) dt – f(uR (–Δx, t; vj–1 tn

tn

n n ) – f(vj–1 ))), and the second space integral is by The two time integrals yield Δt(f(vj+1 n deﬁnition just 2Δx · vj . It therefore follows that

ZΔx

n n n n uR (z, tn+1 ; vj–1 , vj+1 ) dz = 2Δx · vjn+1 = 2Δx · vjn – Δt(f(vj+1 ) – f(vj–1 )) ,

–Δx

which is the desired result. The Godunov scheme could be written as the arithmetic mean of two quantities. In order to tailor the Lax–Friedrichs scheme to this form, we introduce a slight modiﬁcation of the classical Lax–Friedrichs scheme: ” Δt “ ” 1“ n n n n vjn+1 = vj+1 + 2vjn + vj–1 – f(vj+1 ) – f(vj–1 ) , (8.30) 4 2Δx which can be written as ” 1“ – vjn+1 = uj– 1 + u+j+ 1 , 2 2 2

(8.31)

where 1 n n )– (v + vj–1 2 j 1 n = (vj+1 + vjn ) – 2

uj–– 1 = 2

u+j+ 1 2

Δt n )) (f(vjn ) – f(vj–1 Δx Δt n ) – f(vjn )) . (f(vj+1 Δx

(8.32) (8.33)

We call this scheme the modiﬁed Lax–Friedrichs scheme. Due to the modiﬁcation, the numerical viscosity coefﬁcient is now = QmLF j+ 1 2

1 . 2

According to Tadmor’s lemma, we have Δx

vjn+1

1 = 2Δx

Δx

Z2

u – Δx 2

R

n (z, tn+1 ; vj–1 , vjn )

1 dz + 2Δx

Z2 – Δx 2

n uR (z, tn+1 ; vjn , vj+1 ) dz

8.5 The Lax–Friedrichs Scheme

under the CFL condition 1 Δt max | f (vjn )| u . Δx j 2 In complete analogy to the computations performed for the Godunov scheme, we can verify the validity of a discrete entropy inequality S(vjn+1 ) u S(vjn ) –

” Δt “ mLF Gj+ 1 – GmLF 1 j– 2 2 Δx

with numerical entropy ﬂux GmLF = j+ 1 2

” Δx “ ” 1“ n n F(vj+1 ) – F(vjn ) – S(vj+1 ) – S(vjn ) . 2 4Δt

Hence we have shown: Theorem 8.7 Under the CFL condition (8.20), it holds that ” 1 n n f(vjn ) – f(vj–1 ) – (vjn – vj–1 ) 2 “ ” 1 n n u+j+ 1 f(vj+1 ) – f(vjn ) + (vj+1 – vjn ) 2 2 ” Δt “ n S(u–j– 1 ) u S(vjn ) – 2 F(vj ) – GmLF 1 j– 2 2 Δx ” Δt “ n + n mLF S(uj+ 1 ) u S(vj ) + 2 F(vj ) – Gj+ 1 . 2 2 Δx

Δt Δx Δt n = vj – Δx

uj–– 1 = vjn – 2

“

(8.34) (8.35) (8.36) (8.37)

For the case of scalar conservation laws, we have thus derived a complete theory by going back completely to Tadmor’s work. We can now go a step further and develop a theory for all three-point schemes with numerical viscosity coefﬁcient Q satisfying QmLF v Q j+ 1 v QG . j+ 1 j+ 1 2

2

2

This was indeed done, 142) and it leads to a satisfying convergence theory. However, we shall not follow this trail, but instead divert our attention to the Godunov scheme for inviscid ﬂuid ﬂow. Practical experience has revealed that the Godunov scheme is an extremely useful building block for many numerical methods which – at least, if we draw our conclusions from the scalar theory developed above – does not introduce too much numerical viscosity. 142) S. Osher, E. Tadmor: Math. Comp. 50 (1988) 19–51

167

168

8 A Closer Look at Discrete Models

8.6 A Glimpse of Gas Dynamics

The Riemann problem is at the heart of Godunov’s ﬁnite difference scheme. We introduced the Riemann problem in Chapter 4, but we now provide a thorough description of it in the case of one-dimensional inviscid ﬂow described by the Euler equations ∂t V + ∂x f (V ) = 0 with the vector of conserved quantities 0 1 0 1 V1 ρ B C B C V = @V2 A = @ρuA E V3

(8.38)

(8.39)

and with the ﬂux 0

1 ρu B 2 C f (V ) = @ ρu + p A , u(E + p)

(8.40)

as given after Eq. (1.14) in Section 1.1, where we wrote q = ρu. The total energy E is connected with the internal energy e by means of the equation „ « 1 2 (8.41) u +p , E=ρ 2 and we employ an equation of state for ideal gases in the form e = e(p, ρ) = The quantity r γp c= ρ

p . (γ – 1)ρ

(8.42)

(8.43)

is the local velocity of sound; cf. (1.66). Although we have already computed the Jacobian J f (V ) in Section 1.1, we provide three different forms for easy reference here: 0 1 0 1 0 “ ” “ ” 2 B C (3 – γ) VV21 γ – 1 C , (8.44) – 12 (γ – 3) VV21 J f (V ) = B @ “ ”3 “ ”2 “ ”A γV3 – γVV22V3 + (γ – 1) VV21 – 32 (γ – 1) VV21 γ VV21 V1 1 0 1 0 1 0 B (3 – γ)u γ – 1C J f (V ) = @ – 12 (γ – 3)u2 (8.45) A, 3–2γ 2 1 c2 u c2 3 (γ – 2)u – u + γu 2 γ–1 2 γ–1

8.6 A Glimpse of Gas Dynamics

and, introducing the enthalpy H=

p E+p 1 2 = u +e+ , ρ 2 ρ

the Jacobian can again be rewritten to yield 0 0 1 B J f (V ) = @ – 12 (γ – 3)u2 (3 – γ)u ` ´ u 12 (γ – 1)u2 – H H – (γ – 1)u2

1 0 C γ – 1A . γu

(8.46)

The eigenvalues of J f (V ) can be computed to give (cf. (1.15)) λ1 = u – c ,

λ2 = u ,

λ3 = u + c ,

and the corresponding eigenvectors are found to be 0 1 0 1 0 1 1 1 1 B C B C B C K(1) = @ u – c A , K(2) = @ u A , K(3) = @ u + c A . 1 2 H – uc H + uc 2u

(8.47)

(8.48)

For later reference, we rewrite the Euler equations once more – now in terms of the variables density, velocity, and entropy: „ « p + const , s = cv ln ργ where cv is the heat capacity at constant volume. In this formulation, we get the following equation for the pressure: „ « s , (8.49) p = Cργ exp cv where C denotes a constant and the vector of unknowns is 0 1 ρ B C W = @ uA , s while the Euler equations are given in quasilinear form (this is also called the entropy formulation) as ∂t W + A(W)∂x W = 0 with the matrix 0

u

B 2 A(W) = @ cρ 0

ρ u 0

0

(8.50)

1

1 ∂p C ρ ∂s A .

u

(8.51)

169

170

8 A Closer Look at Discrete Models

The eigenvalues of A are given exactly by (8.47), while the eigenvectors are now 0

K(1) s

1 1 B C = @– ρc A , 0

0

K(2) s

1 – ∂p ∂s B C =@ 0 A, 2 c

K(3) s

0 1 1 B C = @ ρc A . 0

(8.52)

Each of the eigenvalues λi can be interpreted as a characteristic speed, and there is a characteristic ﬁeld, called the K(i) -ﬁeld or the i-th characteristic ﬁeld, associated with each speed. A K (i) -ﬁeld is said to be linearly degenerated if, for all states V, the equation ˙ ¸ ∇λi (V ), K (i) = 0

holds. According to Section 3.3, a K (i) -ﬁeld is called genuinely nonlinear if ˙ ¸ ∇λi (V ), K (i) =/ 0

holds. If a discontinuity occurs in the K (i) -ﬁeld, we shall accept it as an admissible discontinuity if the Rankine–Hugoniot condition [ f ] = Si · [V ] holds where [V ] = VR – VL and [ f ] = fR – fL = f (VR ) – f (VL ) denote the jumps across the discontinuity with values VR and VL to the right and the left, respectively. 143) Si is the speed of the discontinuity. An important concept which we will exploit shortly is that of (generalized) Riemann invariants; see Section 3.3. Given an arbitrary one-dimensional quasilinear system ∂t w + A(w)∂x w = 0 where w = (w1 , w2 , w3 ), we consider the K (i) -ﬁeld corresponding to the eigenvalue λi and eigenvector “ ” (i) (i) (i) K (i) = K1 , K2 , K3 .

(Generalized) Riemann invariants are then solutions of the system dw1 (i)

K1

=

dw2 (i)

K2

=

dw3 (i)

K3

.

The Riemann invariants are indeed invariant across waves! 143) See Section 2.3

(8.53)

8.7 Elementary Waves

8.7 Elementary Waves

In a Riemann problem ∂t V + ∂x f (V ) = 0 ( V(x, 0) = V0 (x) =

VL ; VR ;

x<0 x>0

where V = (V1 , V2 , V3 ), we are now looking for similarity solutions, i.e., solutions ` ´ of the form V xt . These solutions are given by four constant states which are seperated by three waves, as shown in Fig. 8.4.

t

λ2

λ1

VL

λ3

VR x

Fig. 8.4 A similarity solution.

In gas dynamics, we know of three different types of waves (see also Section 2.3): 1. Shocks: In a genuinely nonlinear ﬁeld, two constant states VL and VR are connected by an admissible jump discontinuity that additionally satisﬁes the entropy condition, i.e., the Rankine–Hugoniot condition f (VR ) – f (VL ) = Si (VR – VL ) as well as λi (VL ) > Si > λi (VR ) , both hold; see Section 3.3. 2. Contact discontinuities: In a linearly degenerated ﬁeld, two constant states VL and VR are connected by a jump discontinuity if the Rankine–Hugoniot condition f (VR ) – f (VL ) = Si (VR – VL )

171

172

8 A Closer Look at Discrete Models

holds, the Riemann invariants 144) are constant: dw1 (i) K1

=

dw2 (i) K2

=

dw3 (i)

K3

,

and the characteristics are parallel: λi (VL ) = λi (VR ) = Si . 3. Rarefaction waves: In a genuinely nonlinear ﬁeld, two constant states VL and VR are connected by a rarefaction wave if the Riemann invariants are constant: dw1 dw2 dw3 = (i) = (i) , (i) K1 K2 K3 and the characteristics diverge: λi (VL ) < λi (VR ) .

Fig. 8.5 Shock, contact discontinuity, and rarefaction wave (left to right).

Returning to the case of the Euler equations, we look at the eigenvalues (8.47) and the corresponding eigenvectors (8.48), and compute for the K(2) -ﬁeld „ « „ « ∂λ2 ∂λ2 ∂λ2 u 1 , , = – , ,0 ∇λ2 (V ) = ∂V1 ∂V2 ∂V3 ρ ρ so that ˙ ¸ ∇λ2 (V ), K(2) = 0 .

Hence, the K(2) -ﬁeld is linearly degenerate. Analogously, it can easily been shown that the K(1) - and the K(3) -ﬁelds are genuinely nonlinear. Hence, the middle wave corresponding to the wave speed u is always a contact discontinuity. Whether the left and right waves are shock or rarefaction waves depends on the initial values of the Riemann problem. If we consider the Riemann invariants in the conservative variables V, we get dρ d(ρu) dE = = 1 2 , 1 u u 2 from which we conclude dE = 12 u2 dρ, and hence E = 12 u2 ρ + const. Since we know that E = ρ( 21 u2 + e), it follows that the constant is given by ρe, i.e., ρe = const 144) We write wk because these may be different variables from the Vk .

8.7 Elementary Waves

and since e = p/((γ – 1)ρ) for an ideal gas, we conclude const = ρe =

p γ–1

⇒

p = const .

Additionally, we see that d(ρu) = u dρ, so that u = const . We thus conclude that pressure and velocity stay unchanged across a contact discontinuity, while the density may jump. We now consider rarefaction waves. Employing the eigenvectors (8.52) of the Euler equations in entropy formulation (8.50), the Riemann invariants for λ1 = u–c give dρ du ds = c = , 1 –ρ 0 from which we conclude Z c u+ dρ = const , ρ

s = const .

For λ3 = u + c, we analogously obtain Z c dρ = const , s = const . u– ρ

(8.54)

(8.55)

For the pressure in the entropy formulation, we previously (see (8.49)) found that γ

p = Cρ exp

„

s cv

«

.

We can now compute „ « „ « ργ p 1 γp = cv ∂t p – ∂t ρ ∂ t s = cv ∂ t p ργ p ρ ´ cv ` = ∂ t p – c2 ∂ t ρ , p

(8.56)

where the last equation follows from the deﬁnition of the velocity of sound. Analogously, we ﬁnd that ∂x s =

´ cv ` ∂ x p – c2 ∂ x ρ . p

(8.57)

Using the ﬁrst equation in the system of Euler equations in conservative variables, in the case of smooth ﬂow we ﬁnd that ∂t ρ + u∂x ρ + ρ∂x u = 0 ,

(8.58)

173

174

8 A Closer Look at Discrete Models

and using the second equation along with (8.58), we get ∂t u + u∂x u +

1 ∂x p = 0 . ρ

(8.59)

Using (8.58) and (8.59), the third equation in the Euler system (energy equation) yields ∂t p + ρc2 ∂x u + u∂x p = 0 ,

(8.60)

hence ∂t p = –ρc2 ∂x u – u∂x p . Inserting this into (8.56) and exploiting (8.57) yields ∂t s + u∂x s = 0 .

(8.61)

In rarefaction waves, the entropy satisﬁes a linear transport equation; in other words, s = const on the curves dx = u. With this result in hand, we can conclude dt from (8.49) that p = Cργ q for this kind of ﬂow, which is called isentropic (cf. Section 3.1). Hence, c = γp ρ = p γ–1 γCρ is a function of ρ only, and the integrals in (8.54) and (8.55) can be solved to give the Riemann invariants across K(1) - and K(3) -ﬁelds: ) 2c IL (u, c) = u + γ–1 = const λ1 = u – c (8.62) s = const ) 2c IR (u, c) = u – γ–1 = const λ3 = u + c . (8.63) s = const.

It remains for us to discuss shocks which are admissible jump discontinuities in the genuinely nonlinear ﬁelds belonging to K(1) and K(3) , respectively. Consider the K(3) -ﬁeld and a right-facing shock with shock velocity S3 (see Fig. 8.6).

S3 ρ

*

ρR

u*

uR

p

pR

*

Fig. 8.6 Moving shock in the K(3) -ﬁeld.

8.7 Elementary Waves

The right state VR is known in the Riemann problem, while the state to the left of the K(3) -wave is unknown. In order to eliminate the shock speed, we transform the problem into a steady shock. The transformation will not change density and pressure, but rather velocity: uˆ ∗ = u∗ – S3 , uˆ R = uR – S3 , see Fig. 8.7.

S=0 ρ

*

ρR

u^*

u^R

p

pR

*

Fig. 8.7 Steady shock.

The Rankine–Hugoniot condition for shock velocity 0 is ˆR ρ∗ uˆ ∗ = ρR u

(8.64)

ˆ 2R + pR ρ∗ uˆ 2∗ + p∗ = ρR u

(8.65)

ˆ ∗ + p∗ ) = u ˆ R + p∗ ) . ˆ R (E uˆ ∗ (E

(8.66)

With the help of (8.41), the left hand side of (8.66) can be written as „ „ «« 1 2 p∗ uˆ ∗ ρ∗ , uˆ ∗ + e∗ + 2 ρ∗ while the right hand side is „ „ «« 1 2 pR uˆ R ρR . uˆ R + eR + 2 ρR Employing the speciﬁc enthalpy h = e + h∗ = e∗ +

p∗ , ρ∗

hR = eR +

pR . ρR

p ρ

yields (8.67)

With (8.64) and (8.66), we get 1 2 1 uˆ ∗ + h∗ = uˆ 2R + hR . 2 2

(8.68)

175

176

8 A Closer Look at Discrete Models

Inserting (8.64) into (8.65) yields ˆ 2∗ = (ρR uˆ R )ˆuR + pR – p∗ = (ρ∗ uˆ ∗ ) ρ∗ u

ρ∗ uˆ ∗ + pR – p∗ , ρR

from which we derive „ «» – ρR pR – p∗ uˆ 2∗ = . ρ∗ ρR – ρ∗

(8.69)

Analogously, we get „ «» – ρ∗ pR – p∗ uˆ 2R = . ρR ρR – ρ∗

(8.70)

Inserting (8.69) and (8.70) into (8.68) yields » – ρ∗ + ρR 1 h∗ – hR = (p∗ – pR ) . 2 ρ∗ ρR

(8.71)

Now suppose that the internal energy e is given by (8.42). Then it follows from (8.71) with the help of (8.67) that » – ρ∗ – ρR 1 e∗ – eR = (p∗ + pR ) , (8.72) 2 ρ∗ ρR and with (8.42) ρ∗ = ρR

p∗ γ–1 + γ+1 pR γ–1 p∗ · + γ+1 pR

1

.

(8.73)

If we now introduce the Mach number M = MR =

uR , cR

MS =

u a

as

S3 , cR

where MS is called the shock Mach number, then it follows from (8.69) and (8.73) that ρ∗ (γ + 1)(MR – MS )2 = ρR (γ – 1)(MR – MS )2 + 2 p∗ 2γ(MR – MS )2 – (γ – 1) = . pR γ+1

(8.74) (8.75)

From (8.75), we see s

MR – MS = –

γ + 1 p∗ γ – 1 + · , 2γ pR 2γ

and this gives the following expression for the shock speed: s γ + 1 p∗ γ – 1 + · . S3 = uR + cR 2γ pR 2γ

(8.76)

(8.77)

8.7 Elementary Waves

Note that in the case ppR∗ → 1, we get S3 → λ3 = uR + cR , as it should be. Finally, we ﬁnd u∗ from (8.64) to be „ « ρR ρR u∗ = 1 – S3 + uR . (8.78) ρ∗ ρ∗ What have we achieved so far? The equations (8.74), (8.75), (8.78) deﬁne a state (ρ∗ , u∗ , p∗ ) behind a shock in the K(3) -ﬁeld (i.e., behind a right-facing shock) if MS (or S3 ) is given. The computations above are tedious but simple. Nevertheless, in principle we have to go through them again now to get the state to the right of a shock in the K(1) ﬁeld, i.e., a left-facing shock. Since the computations are completely analogous to the previous ones, we will just state the results here. Deﬁning uˆ L = uL – S1 ,

uˆ ∗ = u∗ – S1

and the Mach numbers uL S1 , MS = , ML = cL cL one arrives at ρ∗ = ρL

p∗ pL γ–1 γ+1

+ ·

γ–1 γ+1

p∗ pL

+1

,

(8.79)

and (γ + 1)(ML – MS )2 ρ∗ = . ρL (γ – 1)(ML – MS )2 + 2

(8.80)

Furthermore, p∗ 2γ(ML – MS )2 – (γ – 1) = . pL γ+1

(8.81)

The shock speed S1 follows from either (8.80) or (8.81). Starting from the latter yields s γ + 1 p∗ γ – 1 + ML – MS = · 2γ pL 2γ and hence S1 = uL – cL

s

γ + 1 p∗ γ – 1 + · . 2γ pL 2γ

The velocity u∗ is given by „ « ρL ρL u∗ = 1 – S1 + uL . ρ∗ ρ∗

(8.82)

(8.83)

We have now described the complete theory of gas dynamics within a Riemann problem in detail.

177

178

8 A Closer Look at Discrete Models

8.8 The Complete Solution to the Riemann Problem

We can now obtain the solution to the Riemann problem, which is at the heart of Godunov’s method. The required gas dynamics were all developed above. There are four different cases, which are depicted in Fig. 8.8. It will be convenient to use primitive variables, which we denote by w = (ρ, u, p) , and our general problem statement is given visually in Fig. 8.9.

(b)

(a)

(d)

(c)

Fig. 8.8 Possible wave conﬁgurations.

t u-c

u

w*L w*R

wL

u+c

wR x

Fig. 8.9 Problem statement.

The left and right states wL and wR , respectively, are given. Then there are two unknown states, denoted by w∗L (between the left wave and the contact discontinuity) and w∗R (between the contact discontinuity and the right wave). The left and right waves may be rarefaction waves or shocks.

8.8 The Complete Solution to the Riemann Problem

We know already that p∗L = p∗R =: p∗

(8.84)

u∗L = u∗R =: u∗ .

(8.85)

and

The Riemann problem for the one-dimensional Euler equations is solved if we can compute the values p∗ , u∗ , ρ∗L , ρ∗R . The solution rests on the following result: 145) Theorem 8.8 Computing the pressure and velocity: The pressure p∗ is the solution to the algebraic equation f(p, wL , wR ) := fL (p, wL ) + fR (p, wR ) + Δu

(8.86)

with Δu = uR – uL and

fL (p, wL ) =

8 s > AL > > > (p – pL ) > < p + BL

;

! „ « γ–1 > > p 2γ 2cL > > > –1 :γ – 1 pL 8 s > AR > > > (p – pR ) > < p + BR fR (p, wR ) = ! „ « γ–1 > > p 2γ 2cR > > > –1 : γ–1 pR

2 , (γ + 1)ρL 2 AR = , (γ + 1)ρR AL =

p > pL

(shock)

;

p u pL

(rarefaction)

;

p > pR

(shock)

;

p u pR

,

(8.87)

,

(8.88)

(rarefaction)

9 > > =

BL =

γ–1 γ+1 pL

BR =

> γ–1 > ; γ+1 pR

,

145) We closely follow the presentation given in E.F. Toro: Riemann solvers and numerical methods for ﬂuid dynamics, 2nd ed. Berlin: Springer 1999.

(8.89)

179

180

8 A Closer Look at Discrete Models

and the velocity u∗ is given by u∗ =

´ 1 1` (uL + uR ) + fR (p∗ , wR ) – fL (p∗ , wL ) . 2 2

(8.90)

Proof: The proof proceeds by distinguishing between different possible cases. (I) fL in the case of a left-facing shock with speed SL . In this case, we again transform to a steady shock, deﬁning uˆ L = uL – SL ,

uˆ ∗ = u∗ – SL .

(8.91)

The Rankine–Hugoniot conditions then read as ρL uˆ L = ρ∗L uˆ ∗ ρL uˆ 2L

+ pL =

ρ∗L uˆ 2∗

(8.92) + p∗

(8.93)

ˆ L + pL ) = uˆ ∗ (E ˆ ∗L + p∗ ) . uˆ L (E

(8.94)

Now we deﬁne the mass ﬂux ˆ L = ρ∗L uˆ ∗ . Q L = ρL u

(8.95)

Rewriting (8.93) as (ρL uˆ L )ˆuL + pL = (ρ∗L uˆ ∗ )ˆu∗ + p∗ and using (8.95), we arrive at QL = –

p∗ – pL , uˆ ∗ – uˆ L

(8.96)

but from (8.91) it is seen that uˆ L – uˆ ∗ = uL – u∗ ; based on this and (8.96), we conclude that QL = –

p∗ – pL , u∗ – uL

(8.97)

and therefore u∗ = u L –

p∗ – pL . QL

ˆL = Equation (8.95) yields u Q2L = –

QL , ρL

p∗ – pL . 1 – ρ1L ρ∗L

(8.98) uˆ ∗ =

QL , ρ∗L

which we insert into (8.96) to arrive at (8.99)

8.8 The Complete Solution to the Riemann Problem

Following (8.79), we know that p∗ ! γ–1 γ+1 + pL , γ–1 p∗ γ+1 pL + 1

ρ∗L = ρL

(8.100)

and inserting this into (8.99) gives s

QL =

p∗ + BL , AL

(8.101)

which in turn, and using (8.98), results in u∗ = uL – fL (p∗ , wL ) .

(8.102)

(II) fL in the case of a left-facing rarefaction wave. In this case, the ﬂow is isentropic, and hence p = Cργ .

(8.103)

The constant C is determined at the left state, i.e., pL = CργL , and hence C = Then it follows from (8.103) that r

ρ∗L = ρL

p∗ . pL

pL γ. ρL

(8.104)

We know that the Riemann invariant IL (u, c) is constant across a rarefaction wave: uL +

2cL 2c∗L = u∗ + . γ–1 γ–1

(8.105)

The speed of sound is given by deﬁnition as c∗L =

q

γp∗ ρ∗L ,

and so, using (8.104),

we have v u c∗L = u t

ρL

γp∗ “ ” γ1 = cL p∗ pL

„

p∗ pL

« γ–1 2γ

,

(8.106)

and (8.105) then leads to u∗ = uL – fL (p∗ , wL ) .

(8.107)

Now, turning our attention to right-facing waves, we proceed more rapidly because everything is pretty much analogous to the left-facing cases.

181

182

8 A Closer Look at Discrete Models

(III) fR in the case of a right-facing shock with speed SR . Here, we again transform to a steady shock via uˆ R – SR ,

uˆ ∗ = u∗ – SR

and write down the Rankine–Hugoniot conditions as ˆR ρ∗R uˆ ∗ = ρR u ˆ 2R + pR ρ∗R uˆ 2∗ + p∗ = ρR u ˆ ∗R + p∗ ) = u ˆ R + pR ) . ˆ R (E uˆ ∗ (E The mass ﬂux is now deﬁned as Q R = –ρ∗R uˆ ∗ = –ρR uˆ R . In complete analogy to what we did in the case of a left-facing shock, we arrive at s p∗ + BR QR = AR and u∗ = uR + fR (p∗ , wR ) .

(8.108)

(IV) fR in the case of a right-facing rarefaction wave. This case is also completely analogous to left-facing rarefaction. We ﬁnally arrive at the speed of sound, „

c∗R = cR

p∗ pR

« γ–1 2γ

,

and at the velocity, u∗ = uR + fR (p∗ , wR ) .

(8.109)

In order to conclude, let us state that by eliminating u∗ from (8.98) or (8.107) and (8.108) or (8.109), we get fL (p∗ , wL ) + fR (p∗ , wR ) + Δu = 0. Solving this equation for p∗ gives u∗ depending on the particular case according to (8.98), (8.107), (8.108) or (8.109). One can put all of the cases together in the formula u∗ =

´ 1 1` fR (p∗ , wR ) – fL (p∗ , wL ) . (uL + uR ) + 2 2

Solving (8.86) for the pressure p∗ amounts to solving a nonlinear equation, and iterative methods are required. In the literature, one can ﬁnd useful hints about

8.8 The Complete Solution to the Riemann Problem

the type of methods that are preferentially used. 146) We assume that we now have p∗ as well as u∗ . It now remains to compute ρ∗L and ρ∗R . This can be done according to the particular ﬂow case. 1. A left-facing shock is identiﬁed by p∗ > pL . After having computed p∗ and u∗ , we compute ρ∗L with the help of (8.100) and the shock speed from (8.82). 2. In contrast, a left-facing rarefaction wave is characterized by p∗ u p. After having computed p∗ and u∗ , we compute ρ∗L from the isentropic equation „

ρ∗L = ρL

p∗ pL

« γ1

,

and the speed of sound from „

c∗L = cL

p∗ pL

« γ–1 2γ

.

Coming from the left, the beginning of the rarefaction wave (which we call the head) has a speed of Shead,L = uL – cL . The other end of the rarefaction wave (on the right of it) is the tail of the wave, with speed Stail,L = u∗ – c∗L . If we denote the state at any point within the rarefaction wave by w∗ = (ρ, u, p), and remember the characteristic equation dx x = =u–a dt t as well as the constancy of the Riemann invariant IL (u, c) = const

⇒

uL +

2cL 2c =u+ , γ–1 γ–1

then we can solve both equations simultaneously for u and c. Together with the deﬁnition of the speed of sound as well as the isentropic equation, we get 8 2 “ ” γ–1 γ–1 2 x > > ρ = ρL γ+1 + (γ+1)c (u – ) L > t L > > < “ ” γ–1 2 w∗ = u = γ+1 cL + 2 uL + xt > > > 2γ “ ” γ–1 > > :p = p γ–1 2 x L γ+1 + (γ+1)cL (uL – t ) 146) See the very instructive book by Toro cited previously

.

183

184

8 A Closer Look at Discrete Models

3. A right-facing shock is characterized by p∗ > pR . After having computed u∗ and p∗ , we get p∗ γ–1 ! pR + γ+1 ρ∗R = ρR γ–1 p∗ γ+1 pR + 1 and

s

SR = uR + cR

γ + 1 p∗ γ – 1 + . 2γ pR 2γ

4. A right-facing rarefaction wave is identiﬁed by p∗ u pR . After computing p∗ and u∗ , we compute „ « γ1 p∗ ρ∗R = ρR pR and „

c∗R = cR

p∗ pR

« γ–1 2γ

.

The head of the rarefaction wave is now on the right side, with speed Shead,R = uR + cR , and the tail correspondingly has a speed of Stail,R = u∗ + c∗R . As in the case of the left-facing rarefaction wave, we can compute the state w∗ = (ρ, u, p) anywhere within the wave as 8 2 “ ” γ–1 γ–1 2 x > > ρ = ρR γ+1 – (γ+1)c (u – ) R > t R > > < “ ” γ–1 2 w∗ = u = γ+1 –cR + 2 uR + xt . > > > 2γ “ ” > > γ–1 :p = p γ–1 2 x R γ+1 – (γ+1)cR (uR – t ) We have given the complete solution to the Riemann problem in gas dynamics, and we are now able to compute the solution – at least iteratively up to any desired accuracy – at any point in space-time.

8.9 The Godunov Scheme in Gas Dynamics

In order to solve the Riemann problem ∂t V + ∂x f (V ) = 0 ( V0 (x) = V(x, 0) =

(8.110) VL ;

x<0

VR ;

x>0

(8.111)

8.9 The Godunov Scheme in Gas Dynamics

for the one-dimensional Euler equations, i.e., for 0 1 0 1 ρ ρu B C B C V = @ρuA , f (V ) = @ ρu2 + p A E u(E + p) where E = ρ(u2 /2 + p) and with the ideal gas law e = p/((γ – 1)ρ), we could now use the Godunov recipe as described in (8.17) – (8.19). If we denote a numerical ˜ , we could then write approximation to V by V ˜ n+1 V j

2 Δx 3 Z2 Z0 1 6 R ˜ nj–1 , V ˜ nj ) dz + ˜ nj , V ˜ nj+1 ) dz7 = VR (z, tn+1 ; V 4 V (z, tn+1 ; V 5 , (8.112) Δx 0

– Δx 2

where VR denotes the solution to the Riemann problem in conservative variables. However, while (8.112) is a valid Godunov method, it has a particular disadvantage. We know from the last sections how to solve the Riemann problem completely at any point in space-time that we choose. This is a pointwise solution, but we would have to integrate according to (8.112). Although there is no conceptional problem in choosing a complicated (or simple) quadrature rule, this approach is seldomly used in practice. The Godunov method is in fact based on the following ﬁnite difference form: “ ” ˜ n+1 ˜ nj + Δt gn 1 – gn 1 V =V (8.113) j j– 2 Δx j+ 2 with numerical ﬂux “ ” gnj+ 1 = f VR (0, tn+1 ; Vnj , Vnj+1 ) , 2

(8.114)

and so the exact solution to local Riemann problems is only required at the points xj+ 1 . Another advantage of this implementation comes from the fact that 2 we do not have to work with half the CFL number to avoid wave interactions, as in (8.112). Why have we invested so much effort in the solution of the Riemann problem in order to now arrive at a numerical method? There are at least two important reasons. Although other ﬁnite difference methods are currently in fashion, the Godunov scheme is still important to the development of modern schemes. It can be seen as the grandfather to a whole family of new schemes called Godunov-type methods in which local Riemann problems are solved approximately. The Roe scheme (named after Phil Roe) is a famous scheme of this kind. A book like ours is not the place to develop these classes of schemes further, but the idea of considering the waves in local Riemann problems is relevant here; it is of great importance in numerical methods and will be so even in the future. The second reason for the effort we invested lies in an important pedagogical fact. Anyone starting out in the ﬁelds of modeling in ﬂuid dynamics or numerical modeling should work through the Riemann problem, since this will aid their

185

186

8 A Closer Look at Discrete Models

understanding of trickier situations. There are now attempts to model complicated ﬂows, like multiphase ﬂows, non-Newtonian ﬂuids, ﬂows in the presence of vacuum, low Mach number compressible ﬂows, and so forth. In order to understand not only the phenomenona but also the numerical mathematics required to achieve this, a thorough understanding of the classical Riemann problem is indispensable. We close with some words on the use of the Godunov method. As it stands, the Godunov method (like the Lax–Friedrichs method) is accurate to the ﬁrst order, which is not accurate enough for today’s requirements. Therefore, recovery algorithms (which we shall describe in some detail for ﬁnite volume methods on unstructured grids in Chapter 10) have been developed over the years to increase accuracy. In many of those modern schemes, the Godunov method is still a main building block.

187

9 Discrete Models on Curvilinear Grids 9.1 Mappings

Although there are commercial ﬁnite difference codes on Cartesian meshes where one has to employ sophisticated rules for boundary treatment, the ﬁnite difference methods are usually applied on curvilinear body-ﬁtted grids. Since a wide variety of textbooks and research monographs are available on the subject, we will not go much into the details of the generation of curvilinear grids itself here but instead conﬁne ourselves to modeling aspects. Consider two-dimensional polar coordinates given by ` ´ ` ´ x1 ξ1 , ξ2 = ξ1 cos ξ2 ` ´ ` ´ x2 ξ1 , ξ2 = ξ1 sin ξ2 .

Here we have written ξ1 for the radius and ξ2 for the angle. Consider a circular obstacle with radius R1 in Cartesian (x1 , x2 )-space, as shown in Fig. 9.1. If we want to describe some sort of ﬂow around this obstacle, we need to restrict the far ﬁeld by means of an outer circle, which we assume to have a radius R2 . Using the above transformation, we are now able to address every point between the surface of the obstacle and the farﬁeld circle by prescribing ξ1 ∈ [R1 , R2 ], ξ2 ∈ [0, 2π]. How can we interpret the transformation between so-called physical space (spanned by the Cartesian coordinates) and the parameter space spanned by ξ1 , ξ2 ? Suppose that a slit is inserted along the line AB in Fig. 9.1 and then the space D A C

B D

C

B A

ξ2 ξ1

x2 x1

Fig. 9.1 Polar coordinates.

188

9 Discrete Models on Curvilinear Grids

between the obstacle and the farﬁeld circle is continuously deformed into a rectangular shape. Note that the points A and C are identical in physical space, as are the points B and D. Due to the slit and the continuous transformation into a rectangular shape, these points now appear to be separated, but they are not. In other words, the boundaries CD and AB are periodic boundaries: moving from left to right through the line AB means entering the parameter space again on the left hand side through CD. The reason for considering the transformation above is easy to see. Finite difference methods are constructed on Cartesian grids, but the grid in physical space is not Cartesian. However, the parameter space is Cartesian and so we can now solve the ﬁnite difference equations set up in physical space in the Cartesian parameter space. However, the differential equations describing the ﬂow must be transformed to the parameter domain. Note that we know the transformation relations exactly in the example with the polar coordinates. If we consider an airfoil given as a set of points in the plane (or a set of splines), generally speaking we will not know the transformation exactly. We now turn our attention to describing the general case in two space dimensions. 147) Three-dimensional transformations are much more demanding due to practical issues, but mathematically there is no difference from the 2D case. In general, a coordinate transformation between a parameter space Π 2 and Cartesian space X2 should be a diffeomorphism, i.e., a smooth, invertible mapping. Employing the canonical basis vectors e1 , e2 , we have the Cartesian and parametric coordinates, respectively, given as x = x1 e1 + x2 e2

(9.1)

ξ = ξ e1 + ξ e2 .

(9.2)

1

2

Note that we are dealing with the transformations ( Π 2 → X2 x: ξ → x(ξ) ( X2 → Π 2 ξ: x → ξ(x) which are inverse to each other. Writing ` ´ ` ` ´ ` ´´ x ξ1 , ξ2 = x ξ1 x1 , x2 , ξ2 x1 , x2

and differentiating, we get, with the help of (9.1), ∂x ∂ξ1 ∂x ∂ξ2 ∂x = e1 = 1 1 + 2 1 1 ∂x ∂ξ ∂x ∂ξ ∂x 147) A beautiful treatment is given in the book by V.D. Liseikin: Grid generation methods. Berlin: Springer 1999.

9.1 Mappings

∂x ∂x ∂ξ1 ∂x ∂ξ2 = e = + . 2 ∂x2 ∂ξ1 ∂x2 ∂ξ2 ∂x2 We can write each of the equations in the form of a matrix-vector product as ∂ξ , i = 1, 2 ∂xi with the help of the Jacobian matrix 2 1 3 ∂x ∂x1 6 ∂ξ1 ∂ξ2 7 6 7 J := 6 7. 4 ∂x2 ∂x2 5 ei = J ·

(9.3)

(9.4)

∂ξ1 ∂ξ2 Combining both Eqs. (9.3) into one, we arrive at the matrix equation 2 1 3 ∂ξ ∂ξ1 6 ∂x1 ∂x2 7 6 7 I= J·6 (9.5) 7, 4 ∂ξ2 ∂ξ2 5 ∂x1 ∂x2 where I denotes the two-dimensional unit matrix. We now turn our attention to ` ´ ` ` ´ ` ´´ ξ x1 , x2 = ξ x1 ξ1 , ξ2 , x2 ξ1 , ξ2 , which, after differentiation and with the help of (9.2), yields ∂ξ ∂x1 ∂ξ ∂x2 ∂ξ = e = + , i ∂ξi ∂x1 ∂ξi ∂x2 ∂ξi or, again in the form of a matrix equation, 2 1 3 ∂x ∂x1 6 ∂ξ1 ∂ξ2 7 6 7 I= K·6 7 4 ∂x2 ∂x2 5

(9.6)

∂ξ1 ∂ξ2 where K is the matrix 2 1 3 ∂ξ ∂ξ1 6 1 7 6 ∂x ∂x2 7 K=6 7. 4 ∂ξ2 ∂ξ2 5 ∂x1 ∂x2 Comparing this matrix K with (9.5) yields the ﬁrst fundamental result, K = J –1 .

(9.7)

Hence, if det J is the Jacobian determinant, then det K = 1/det J. The elements of the matrices J and K = J–1 are related through 1 ∂x3–j ∂ξi = (–1)i+j , j ∂x det J ∂ξ3–i ∂ξ3–j ∂xi = (–1)i+j (det J ) 3–i , j ∂ξ ∂x

i, j = 1, 2 i, j = 1, 2

which can easily be computed from (9.7).

189

190

9 Discrete Models on Curvilinear Grids

9.2 Transformation Relations

The value x(ξ) gives a point in physical space corresponding to a given point ξ in ∂x parameter space. Geometrically, each of the vectors ∂ξ i is a tangent vector to the j grid line ξ = const, j =/ i. These vectors are called the covariant basis vectors. It is ∂x easily seen that det J is the volume of the parallelogram spanned by ∂ξ i , i = 1, 2. In “ i ” i ∂ξ ∂ξ i contrast, the vectors ∇ξ = ∂x1 , ∂x2 are vectors normal to the lines of constant ξi . They are called the contravariant vectors. Suppose now that two linearly independent vectors a1 , a2 ∈ R2 are given as well as a further vector b ∈ R2 with components b1 , b2 with respect to the canonical basis {e1 , e2 }. If we want to represent the vector b as a linear combination of the ai , then we recall from linear algebra that 2 2 X X

b=

˙ ¸ aij b, aj ai

(9.8)

i=1 j=1

` ´ where aij are elements of a matrix aij which is the inverse of the matrix (aij ) deﬁned through ˙ ¸ aij = ai , aj ,

i, j = 1, 2 .

Expanding the vector b in terms of the basis b=b

1

n

∂x ∂ξi

o , i = 1, 2, i.e.,

∂x 2 ∂x +b , 1 ∂ξ ∂ξ2 i

then the components b are called the contravariant components of b. Taking the ∂x vectors ai in (9.8) to be ai = ∂ξ i , i = 1, 2, then we obtain b=

XX i

XX ˙ ¸ a b, aj ai = aij ij

j

i

j

ﬁ

∂x b, j ∂ξ

ﬂ

∂x ∂ξi

» ﬁ ﬂ ﬁ ﬂ– ∂x ∂x ∂x = a11 b, 1 + a12 b, 2 ∂ξ ∂ξ ∂ξ1 {z } | 1

=b

» ﬁ ﬂ ﬁ ﬂ– ∂x ∂x ∂x + a21 b, 1 + a22 b, 2 , ∂ξ ∂ξ ∂ξ2 | {z } 2

=b

i.e., i

b =

X j

ﬁ ﬂ ∂x aij b, j . ∂ξ

(9.9)

9.2 Transformation Relations

Since aij =

X ∂xk ∂xk , ∂ξi ∂ξj k

it follows that aij =

X ∂ξi ∂ξj , ∂xk ∂xk k

and hence, from (9.9), that X X ∂ξi ∂ξj ﬁ ∂x ﬂ ˙ ¸ b = b, j = b, ∇ξi , k ∂xk ∂x ∂ξ j i

(9.10)

k

due to the inverse character of reads b=

∂ξi ∂xk and . Hence, in this case, the relation (9.8) k ∂ξi ∂x

X˙ ¸ ∂x b, ∇ξi , ∂ξi i

(9.11)

and in the special case b = ∇ξi , we see that ∇ξi =

X X ∂ξi ∂ξk ∂x , ∂xj ∂xj ∂ξk j

i = 1, 2 .

(9.12)

k

In a completely analogous manner, we can compute the covariant components bi of b in the expansion b = b1 ∇ξ1 + b2 ∇ξ2 as ﬁ

bi =

b,

∂x ∂ξi

ﬂ

.

Consequently, b can also be represented as b=

X ﬁ ∂x ﬂ b, i ∇ξi , ∂ξ

(9.13)

i

and if we choose the special case b = ∂x X X ∂xj ∂xj = ∇ξk , i ∂ξk ∂ξi ∂ξ j k

∂x , we get ∂ξi i = 1, 2 .

(9.14)

191

192

9 Discrete Models on Curvilinear Grids

9.3 Metric Tensors

As we have seen, all geometric information about coordinate transformations between parameter space and physical space is already contained in the basis vectors ∂xi . The inner products of these vectors comprise the elements ∂ξj ﬁ ﬂ ∂x ∂x , (9.15) gij = ∂ξi ∂ξj of the covariant metric tensor G = (gi,j ) ,

i, j = 1, 2 .

This tensor is one of the fundamental entities in the differential geometry of coordinate transformations. Comparing (9.4) with (9.15), we immediately see that G = J · JT and hence g := det G = (det J )2 . In analogy, there is also a contravariant metric tensor (gij ) which is deﬁned to be the inverse of the covariant metric tensor: ( X 0; i =/ k ij k g gjk = δi = , i, k = 1, 2 . 1; i = k j From this relation, we readily compute ˙ ¸ gij = ∇ξi , ∇ξj and det(gij ) =

1 . g

Convenient formulae can now be derived which are quite useful. We give only a few of them and direct the reader to the literature. 148) The following relations are readily apparent: gij = (–1)i+j ·

1 · g3–i 3–j , g

i, j = 1, 2

148) A book that is almost indispensable for practical work is J.F. Thompson, Z.U.A. Warsi, C.W. Mastin: Numerical grid generation: foundations and applications. Amsterdam: North-Holland 1985.

9.4 Transforming Conservation Laws

gij = (–1)i+j · g · g3–i 3–j ,

i, j = 1, 2 .

Looking back at (9.12) and (9.14), we see that our transformation relations between the two types of basis vectors can be written as ∂x X = gik ∇ξk , ∂ξi

i = 1, 2

k

∇ξi =

X k

gik

∂x , ∂ξk

i = 1, 2 .

9.4 Transforming Conservation Laws

We are now ready to adress the problem of transforming a given conservation law in physical space to one in parameter space. The derivatives of the Jacobian deter√ minant det J = g form a key ingredient. Since det J =

∂x1 ∂x2 ∂x1 ∂x2 – , ∂ξ1 ∂ξ2 ∂ξ2 ∂ξ1

∂ is quite involved in terms of the number of ∂ξk terms which must be written down, but there are no conceptual difﬁculties. The result is the computation of the derivative

∂ ∂x det J = (det J )∇x · k , k ∂ξ ∂ξ where ∇x · denotes the divergence with respect to Cartesian coordinates. From this relation, we can derive the important formula 149) « X ∂ „ ∂ξj (det J ) =0. (9.16) ∂ξj ∂x j Consider a function x2 (x1 , x2 ) → f (x1 , x2 ) ∈ R deﬁned in physical space. In order to compute a representation of the ﬁrst derivatives of f in parameter space, we view f as f (x1 (ξ1 , ξ2 ), x2 (ξ1 , ξ2 )), as above, and take derivatives. From what we derived above, it is clear that ∇ξ f = J · ∇ x f , i.e., ∇x f = J–1 · ∇ξ f . 149) For a proof see, e.g., V.D. Liseikin: Grid generation methods. Berlin: Springer 1999, 46ff.

193

194

9 Discrete Models on Curvilinear Grids

Now, since 2

J–1

3 ∂x2 ∂x1 – 7 1 6 6 ∂ξ2 ∂ξ2 7 = 6 7, det J 4 ∂x2 ∂x1 5 – 1 ∂ξ ∂ξ1

we obtain the following for the partial derivatives: ∂f 1 = ∂x1 det J 1 ∂f = ∂x2 det J

„ „

∂x2 ∂f ∂x1 ∂f – ∂ξ2 ∂ξ1 ∂ξ2 ∂ξ2 –

«

∂x2 ∂f ∂x1 ∂f + 1 2 1 1 ∂ξ ∂ξ ∂ξ ∂ξ

(9.17) «

.

(9.18)

The key ingredient of the transformation of conservation laws is the following theorem. Theorem 9.1

∂f 1 X ∂ = i ∂x det J j ∂ξj

„

(det J )

« ∂ξj f , ∂xi

i = 1, 2 .

Proof: We compare the formulae (9.17) and (9.18) with the formula given in the lemma. Starting from the latter, we get „ « ∂ξj 1 X ∂ (det J ) f det J j ∂ξj ∂xi » „ « „ «– ∂ 1 ∂ξ1 ∂ ∂ξ2 = (det J ) f + (det J ) f det J ∂ξ1 ∂xi ∂ξ2 ∂xi » „ 1« 1 ∂ 1 ∂ξ ∂ξ ∂f = (det J ) i f + (det J ) i 1 det J ∂ξ1 ∂x ∂x ∂ξ „ – 2« 2 ∂ ∂ξ ∂ξ ∂f + 2 (det J ) i f + (det J ) i 2 . ∂ξ ∂x ∂x ∂ξ

Using (9.16), we see that the sum of the ﬁrst and the third terms within the square brackets vanishes. Hence, we end up with 1 X ∂ det J j ∂ξj

„

(det J )

∂ξj f ∂xi

«

=

∂ξ1 ∂f ∂ξ2 ∂f + . i 1 ∂x ∂ξ ∂xi ∂ξ2

If we now replace the contravariant vectors with covariant ones according to the formulae we have already derived, we obtain (9.17) and (9.18) directly. We can now give two examples.

9.4 Transforming Conservation Laws

Example Consider the Laplace equation Δf = 0 , which is obviously a conservation law, since Δf = ∇ · (∇f ). Since Δf =

X ∂ ∂f , ∂xi ∂xi i

we can replace the partial derivatives by means of Theorem 9.1 to yield ∂ ∂f 1 X ∂ = i i ∂x ∂x det J j ∂ξj

(det J )

∂ξj X 1 ∂ · ∂xi det J ∂ξk k

= ... 1 X ∂ = det J j ∂ξj

(det J )

X k

∂f g ∂ξk kj

(det J )

∂ξk f ∂xi

!!

!

.

Thus, the Laplace equation in the parameter space reads ! X 1 X ∂ kj ∂f (det J ) g =0. det J j ∂ξj ∂ξk k

Example In our mathematical models of ﬂuid ﬂow, we frequently encounter derivatives of vector-valued ﬂux functions in the form ∇ · f (u) =

X ∂f i . ∂xi i

Using (9.16), we arrive at ∇ · f (u) =

” 1 X ∂ “ j (det J · f (u) , j det J j ∂ξ

(9.19)

j

in which f denotes the j-th contravariant component of the vector f, i.e., j

f (u) =

X i

f i (u)

∂ξj . ∂xi

Hence, the divergent part of a conservation law given in physical space transforms into (9.19) in parameter space. Note that this transformed term is again in conservation form.

195

196

9 Discrete Models on Curvilinear Grids

We shall not describe second-order or higher derivatives, since the formulae follow along the lines developed here. Many books contain ready-to-use formulae 150) so that it is not necessary to work through tedious computations to arrive at transformed equations. Care must be taken to transform a conservation law in the physical domain into a conservation law in parameter space. It is much easier to transform a conservation law into an equation that does not possess the property of conservation.

9.5 Good Practice

We have expended quite some effort to show how differential equations can be transformed from physical space, in which the ﬂow actually takes place, to parameter space, where ﬁnite difference methods can work. We have not, however, said a single word about the actual construction of the transformation, which is equivalent to the process of grid generation. In discretizing the transformed conservation laws, we now have to discretize the metric terms too. In practice, one always thinks of the parameter domain as being discretized in the following way: ˆ ˜ ξi ∈ 1, 2, 3, . . . , Ni Δξi = 1 . Hence, the discretized parameter space consists of integer values of the parameters with a spacing of 1. Depending on the computer language chosen, the parameters may take values ξi ∈ 0, 1, 2, . . . , Ni –1, so that the numbering is somewhat arbitrary. This situation has already been shown in Fig. 9.1, where the grid lines and their corresponding straight lines in the parameter domain are depicted. For the metric ∂x terms, the most important of which are the derivatives j , we can now choose any ∂ξ ﬁnite difference of choice. The most popular one used in production codes is the simple difference ` ´ ` ´ ∂x W x ξ1 + 1, ξ2 – x ξ1 , ξ2 1 ∂ξ ` ´ ` ´ ∂x W x ξ1 , ξ2 + 1 – x ξ1 , ξ2 . 2 ∂ξ

Since the discrete parameter values are integers, it makes sense to write x(i, j) in` ´ stead of x ξ1 , ξ2 . The ﬁrst step in the process of grid generation is to choose the grid topology. As an example, Fig. 9.2 shows a C-type grid around an airfoil-like obstacle, where the ﬂow is assumed to come from the left. 150) See, e.g., J.F. Thompson, Z.U.A. Warsi, C.W. Mastin: Numerical grid generation: foundations and applications. Amsterdam: North-Holland 1985

9.5 Good Practice

H G

D B

C

i

E A

j F

Fig. 9.2 C-type grid.

The corresponding parameter space is given in Fig. 9.3. It can be thought of as the physical space slit along the line AB and then continuously transformed into a rectangle. Note that the parts AB and DE of the parameter line j = 1 are logically identical. The surface of the obstacle corresponds to the line segment BCD, and the farﬁeld is given by FGH. Note that x(1, 1) would correspond to A, x(4, 1) would correspond to B, and x(3, 1) to F. G

F

j A

i

B

H

......

........

......

........ C

D

E

Fig. 9.3 C-type parameter space.

In contrast to the C-type topology, we could also have chosen the O-type grid, as shown in Fig. 9.4.

C i B j

D A Fig. 9.4 O-type grid.

This type of grid topology is just a slight generalization of polar coordinates. Yet another topology is given by the H-type grid; see Fig. 9.5. Here, the grid consists of two nearly Cartesian parts which ﬁt logically together at the slit line ABCDEFGH. H-type grids are only used rarely for airfoils, because the grid quality near the leading and trailing edges of the airfoil is much worse than in the C- or O-type grids. By the grid quality, we mean the angles of the grid lines of the i-family with those of the j-family. A grid would be optimal if it was

197

198

9 Discrete Models on Curvilinear Grids

j E Di

F C

G B

H A

j i

Fig. 9.5 H-type grid.

orthogonal everywhere, but this condition is seldom met. However, we should always try to get as close as possible to orthogonality, at least in regions sensible to the ﬂow. To describe only some of the hundreds of algorithms available for grid generation in detail would already be far beyond the goals of this book. The ﬁrst step in grid generation is always to prescribe points on the surfaces of the obstacles in the ﬂow and on the farﬁeld. This is usually done with the help of spline functions or other interpolation techniques. Techniques used to generate the ﬁeld grid, i.e., the grid points between the obstacle and farﬁeld, can be divided into algebraic and analytic methods of grid generation. Interpolation techniques belong to the algebraic category. Stretching methods that allow a prescribed point density to be distributed on a line or plane can also be found in this category. Analytic methods compute grid points as solutions of partial differential equations or variational problems. The simplest partial differential equation used is the Laplace equation Δξ = 0 for the parametric coordinates. Since we cannot solve for functions ξ(x) but we can for functions x(ξ) (these are our grid points), we need to transform the above equation along the lines outlined above to get 3 2 X X i=1 j=1

gij

∂2x =0. ∂ξi ∂ξj

In terms of covariant metric coefﬁcients instead of contravariant ones, we get g22

∂2x 12

∂ξ

+ g11

∂2x ∂2x – 2g12 1 2 = 0 . 2 ∂ξ ∂ξ ∂ξ2

(9.20)

To solve for the position of the grid points, we have to prescribe an initial grid; this is usually derived via an algebraic method, i.e., interpolation. The second derivative that occurs is replaced by a ﬁnite difference, and the metric coefﬁcients gij are also numerically computed by means of ﬁnite differences. Equation (9.20) is then a linear system of equations for the grid point positions x(i, j), which can be solved by an iterative method.

9.5 Good Practice

In order to analyze the properties of the grids generated by this system we consider the following case. We assume that ξ2 = 1 describes the surface line of a twodimensional convex body (consider it a circle), and ξ2 > 1 are grid lines around that body but farther away. The ξ2 = const family is assumed to be a solution of the Laplace equation Δξ2 = 0 , and that implicit function theorem allows the representation of ξ2 (x1 , x2 ) = const locally in the form x2 = f (x1 ). Furthermore, we assume that (x2 ) (0) = 0

(9.21)

holds along the x2 -axis, i.e., that every grid line passes through the x2 -axis with a horizontal tangent. Let the grid line ξ2 (x1 , x2 ) = c2 pass through the point (0, x2∗ ), another line ξ2 (x1 , x2 ) = c1 pass through (0, x2∗ – Δx2 ), and a third one ξ2 (x1 , x2 ) = c3 pass through (0, x2∗ + Δx2 ), so that c1 < c2 < c3 . The implicitly deﬁned curve ξ2 (x1 , x2 ) = const has a curvature given by

κ=

∂ 2 ξ2 2 ∂x1

“

∂ξ2 ∂x2

”2

ξ ∂ξ ∂ξ ∂ξ – 2 ∂x∂1 ∂x 2 ∂x1 ∂x2 + ∂x2 2 s„ “ 2 ”2 “ 2 ”2 «3 ∂ξ ∂ξ + ∂x 2 ∂x1 2 2

2

2

2

“

∂ξ2 ∂x1

”2

.

(9.22)

Since the derivative in the explicit form x2 = f (x1 ) is given by ∂ξ2

(x2 ) = – ∂x , ∂ξ2 1

∂x2

= 0 at every point on the x2 -axis. we conclude from assumption (9.21) that ∂ξ ∂x1 Substituting back into (9.22), we get the following for the curvature κ0 of the grid lines at x1 = 0: “ 2 ”2 2 2 2

∂ ξ 2 ∂x1

κ0 = r“

∂ξ ∂x2

∂ξ2 ∂x2

”6 .

Due to the convexity of the surface line, it holds that κ0 > 0 and therefore that ∂ 2 ξ2 ∂x1 2

>0.

However, since Δξ2 = 0 must be satisﬁed, we then obtain ∂ 2 ξ2 <0. ∂x2 2

(9.23)

199

200

9 Discrete Models on Curvilinear Grids

Replacing this last derivative by a ﬁnite difference of second order yields ` ´ ` ´ ` ´ ∂ 2 ξ2 ξ2 0, x2∗ + Δx2 – 2ξ2 0, x2∗ + ξ2 0, x2∗ – Δx2 W , (Δx2 )2 ∂x2 2 and it follows from (9.23) that ` ´ ` ´ ` ´ ` ´ ξ2 0, x2∗ + Δx2 – ξ2 0, x2∗ < ξ2 0, x2∗ – ξ2 0, x2∗ – Δx2 , or, due to the numbering of the grid lines, c3 – c 2 < c 2 – c 1 . Hence, in the case of a convex body, the number of grid lines is larger in the interˆ ˜ ˆ ˜ val x2∗ – Δx2 , x2∗ than in the interval x2∗ , x2∗ + Δx2 . Had we considered a concave surface, we would have ended up with c3 – c2 > c 2 – c1 . Hence we can state the following: Grid lines generated with system (9.20) move towards convex boundaries and tend to move away from concave ones. A typical grid exhibiting the behavior demonstrated above can be seen in Fig. 9.6.

Fig. 9.6 Laplace grid around curved shape.

Equation (9.20) does not allow the user any kind of control over the grid line position. In order to remedy this, we can introduce two control functions or source terms Q1 and Q2 and solve the Poisson-type equation ! 1 g11 Q1 . Δξ = (det J )2 g22 Q2 „

« Q1 = Q instead, but practice Q2 has shown that the contravariant metric coefﬁcients used as factors as well as the square of the Jacobian determinant of the transformation add stability and robustness. In parameter space, this Poisson-type system reads „ 2 « „ 2 « ∂ x ∂ x ∂2x 1 ∂x 2 ∂x g22 + Q + g + Q – 2g12 1 2 = 0 . (9.24) 11 2 2 1 2 2 1 ∂ξ ∂ξ ∂ξ ∂ξ ∂ξ ∂ξ

We could have used the simpler system Δξ =

9.5 Good Practice

If we now follow the analysis we presented for the Laplace equation for the Poisson equation Δξ = Q instead, we immediately see that negative values of Qi force the grid lines of the ξi = const family towards decreasing values, and that positive values of Qi force the grid lines of the ξi = const family towards increasing values. This principle is shown in Fig. 9.7. ξ2

Q2 < 0

2

ξ

ξ1

Q1 < 0

1

ξ

Fig. 9.7 Inﬂuence of the source terms Q1 and Q2 .

There are several strategies for controlling the source terms Qi , and we refer the reader to the literature. 151) As an example, we discuss one of the simplest but most successful strategies, which is called the method of Hilgenstock. 152) It is obvious from the discussion above that, at the boundaries of the physical domain, the angle of the outgoing grid lines as well as the spacing of the ﬁrst grid line from the boundary can be inﬂuenced by the source terms. Let ξ2 = 1 be the line of the boundary of interest (the airfoil, etc.). The direction ξ2 is associated with the source term Q2 , and so Q2 is responsible for the spacing between the grid lines of this family. Let dr be the required spacing and ˛ ` ´ ` ´˛ d = ˛x ξ1 , ξ2 + 1 – x ξ1 , ξ2 ˛

be the true spacing, which can be computed from the initial grid. If the difference dr – d vanishes, then no change in Q2 is necessary; if dr – d > 0, then the spacing of the given grid is too small and Q2 has to grow. If dr – d < 0, then the spacing of the grid is too large and Q2 must decrease. The simplest function satisfying these requirements is Q2(k+1) = Q2(k) + (dr – d) , where (k) denotes an iteration level. In practice, this function can change its values drastically during the ﬁrst few iterations, so that a better choice is Q2(k+1) = Q2(k) + arctan(dr – d) 151) A good overview is provided in the book by Thomson, Warsi and Mastin cited before, but see also T. Sonar: Grid generation using elliptic partial differential equations (Tech. Rep. 89-15). Braunschweig: DFVLR 1989.

(9.25) 152) A. Hilgenstock: A method for the elliptic generation of three-dimensional grids with full boundary control (Int. Rep. 221-87 A 09). Braunschweig: DFVLR 1987

201

202

9 Discrete Models on Curvilinear Grids

instead. Now consider a required angle αr of the ξ1 = const family at the boundary. The true angle can be computed from the given grid as 0D

∂x ∂x , ∂ξ1 ∂ξ2

E1

α = arccos @ ˛˛ ˛˛ ˛˛ ˛˛ A . ∂x ∂x ˛ ∂ξ 1 ˛ ˛ ∂ξ2 ˛ If αr – α > 0, then the actual angle is too small and Q1 must decrease; if αr – α < 0, then the angle is too large and Q1 must increase. The simplest function implementing this is Q1(k+1) = Q1(k) – (αr – α), but the same remarks as above apply and Q1(k+1) = Q1(k) – arctan(αr – α)

(9.26)

is a better choice. The typical grid generation cycle can then described as follows: 1. Let an initial grid be provided 2. Set Q1 = Q2 = 0 everywhere 3. Outer iteration: Set k = 0 (a) Compute d and α at the boundary in the grid (b) Compute (9.25) and (9.26) (c) Inner iteration i. Solve (9.24) numerically via an iterative method ii. The result is a new grid given by its points x(i, j) (d) Set k = k + 1 (e) Continue outer iteration A grid generated by this approach is shown in Fig. 9.8. The required angle at the boundary was chosen to be 90◦ . Working with Laplace- or Poisson-type equations is known as elliptic grid generation, and has proven to be enormously successful in the past thirty years. These elliptic equations not only ﬁt the boundary conditions in grid generation (prescribed point positions at the boundaries or even prescribed angles with the boundary) perfectly, but they permit very good control over the grid and also result in very smooth grids. There are, however, successful attempts to use parabolic or hyperbolic systems. In such cases, the point distribution must only be prescribed at the surface of the obstacle in the ﬂow (this is the initial condition for the hyperbolic system), and the ﬁeld points are then computed by marching away from the obstacle. It is not possible to prescribe the farﬁeld. In general, grid generation is still an art more than a science, and marvellous two- and three-dimensional grids can be found in the literature.

9.6 Remarks Concerning Adaptation

Fig. 9.8 Grid generated by an elliptic system with source term control.

9.6 Remarks Concerning Adaptation

The source terms Q1 and Q2 can be used not only to control grid spacing and angles at the boundary but also within the ﬁeld. Hence, grid adaptivity can be achieved by carefully steering the source terms. Suppose a function f on a one-dimensional grid gives an indication of some error, w = f (x) . The basic idea of adaptation is that the error should be equidistributed over the domain of w, i.e., w(x) · Δx = const . Upon introducing x as a function of ξ, this can be rewritten as w(x(ξ))(x(ξ + Δξ) – x(ξ)) = const , or it can be recast as w(x(ξ))

dx = const . dξ

If we differentiate this equation with respect to ξ, we get 153) w

d2 w dw dx + =0. dξ2 dξ dξ

(9.27)

In the case of one dimension, our grid generation system (9.24) boils down to d2 x dx +Q =0, dξ2 dξ 153) Note that there is now only one ξ, and d2 so dξ 2 means the second derivative with respect to ξ!

(9.28)

203

204

9 Discrete Models on Curvilinear Grids

and if we compare this equation with (9.27), we see that Q=–

d2 x dξ2 dx dξ

.

(9.29)

Now, from (9.27) we see that d2 x dξ2 dx dξ

=–

dw dξ

w

,

and inserting this into (9.29) results in Q=

dw dξ

w

.

We have thus shown that, if we have an error function w, a grid can be generated by solving (9.28) with the above source term on which the error is equidistributed (which is more dense where the error is high). We generalize this idea to two dimensions by simply setting the source terms to Qi =

dw dξi

w

,

i = 1, 2 .

(9.30)

A very simple but efﬁcient error function for applications in compressible ﬂow is the function w = 1 + |∇p| , in which the constant 1 serves to avoid a vanishing w if the pressure gradient is close to zero. In practice, a good adaptation strategy is given by the following recipe. Keep the sources in the grid from the grid generation process, and call these sources Qiold . Compute the error function w and call the sources (9.30) Qiadap . Then, in the outer iteration of the grid generation process, compute Qi = Qiold + c · Qiadap and generate a grid with these new source terms. The constant c is a damping factor u1 that is used to keep the inner iteration stable.

205

10 Finite Volume Models 10.1 Difference Methods on Unstructured Grids

We have already discussed ﬁnite volume methods in Section 7.8, and we arrived at Eq. (7.39): Z D E XZ 1 {un+1 (x, y) – un (x, y)} dσi + f (un ), ni,k ds = O(Δt) (10.1) Δt σi ∂σi,k k∈Ii

where ni,k denotes the outer unit normal vector at the edge ∂σi,k , and where we have used a simple forward difference in time. 154) Writing Z 1 u0 (x, y) dσi vi0 = |σi | σi for the cell average of the initial functions, neglecting the error term in (10.1), and writing si,k for the length of the edge ∂si,k resulted in (7.40): vin+1 = vin –

E Δt X D v f (˜ vi,k ), ni,k si,k . |σi | k∈Ii

Evaluating the inner product on certain polygonal cells σi will result in a ﬁnite difference scheme due to the signs of the components of the normal vector. Since the kind of volumes σi used is arbitrary – the σi may be a quadrilateral, a triangle or any other polygonal shape – it is natural from a ﬁnite difference point of view to call ﬁnite volume schemes ﬁnite difference methods on unstructured grids, and they have been called this in some of the literature. 155) In 2007, a survey work on ﬁnite volume methods was published 156) where the authors exploited a slightly different point of view: ﬁnite volume methods can also be interpreted as a kind of discontinuous Galerkin FEM. 154) This by no means necessary; any time-stepping scheme can be used. 155) See, for example, B. Heinrich: Finite difference methods on irregular networks. Stuttgart: Birkhäuser 1987

156) K.W. Morton, T. Sonar: Acta Num. 16 (2007) 155–238

206

10 Finite Volume Models

For the sake of simplicity, we will employ the ﬁnite volume approach for regular triangulations in the plane. In general, there are three degrees of freedom for the type of discretization when a ﬁnite volume method is being set up, namely a cellcenter approach, a cell-vertex approach, and a node-centered approach.

Fig. 10.1 Cell-centered and cell-vertex approach.

The approaches differ in terms of the locations of the unknowns, and hence lead to different cells or control volumes σi . We have found the node-centered formulation to give the most pleasing results. In this approach, we consider the barycentric subdivision of triangles in wich the barycenter of the triangle itself is connected to the barycenters (i.e., midpoints) of the edges via straight lines.

Fig. 10.2 Barycentric subdivision and the resulting primary and secondary grids.

The barycentric subdivisions build a new cell around each node: the control volume σi belonging to node i of the triangulation. While the triangulation is called the primary grid, we call the set of all control volumes the secondary grid. The geometry between two control volumes, which are also called boxes, is shown in Fig. 10.3. Due to the barycentric subdivision of the triangles, the box boundary now consists of two parts ∂s1ik and ∂s2ik with lengths s1ik and s2ik , respectively, which give rise ik to two outer (with respect to σi ) unit normal vectors nik 1 and n2 . Hence, we can rewrite (10.1) in the form d 1 dt |σi |

Z

2

σi

u dσi = –

1 XX |σi | =1 k∈Ii

Z

D ∂σik

E f (u), nik ds ,

(10.2)

10.1 Difference Methods on Unstructured Grids

sik2

n1ik

sik1 i

k

n 2ik

c2

c1 j

Fig. 10.3 Geometry between two boxes.

where we have allowed for a more general time-stepping scheme. It is immediately apparent from this formulation that the ﬁnite volume methods are numerical schemes for computing cell averages over the boxes. In order to give us some freedom in terms of the accuracy of the resulting scheme, we now introduce a parametrization of each of the boundary segments ∂σik , = 1, 2 in the form „ « „ « 1 1 ζ 1 [–1, 1] ζ → xik (ζ) = c + (xi + xk ) + (xi + xk ) – c , 2 2 2 2 where c1 and c2 are the coordinates of the barycenters of the two adjacent triangles sharing edge ik, and xi , xk denote the coordinates of the points i and k of the triangulation, respectively. Aided by this parametrization, we are now able to introduce Gaussian quadrature points on ∂σik , so that (10.2) now reads as "n # Z 2 G D ““ ”” E d 1 1 X X sik X ik 2nG u dσi = – ω f u xik (ζν ), t , n + O(h ) dt |σi | σi |σi | 2 ν=1 ν =1 k∈Ii

(10.3) where nG denotes the number of Gaussian points, ων corresponds to the weights of the quadrature rule, and the error includes a typical length measure h of the triangulation involved. We now introduce a numerical ﬂux function, as we did in the derivation of (7.44). In that case we included sik in the deﬁnition of the numerical ﬂux gi,k because this was convenient in the derivation of a low-order ﬁnite volume scheme employing only one Gaussian point. Here we redeﬁne our deﬁnition, and we want to introduce a numerical ﬂux in the form (ul , ur ; n) → g(ul , ur ; n) , which is required to satisfy the consistency condition g(w, w; n) = f (w), n . When compared to (7.43), it is obvious that only sik is excluded from our redeﬁnition. Introducing a numerical ﬂux function of this kind into (10.3) yields Z d 1 u dσi dt |σi | σi

207

208

10 Finite Volume Models

=–

"n # 2 G “ “ ” “ ” ” 1 X X sik X 2nG ων g u xik (ζν ), t , u xik (ζν ), t ; nik + O(h ) . |σi | 2 ν=1 =1 k∈Ii

(10.4)

10.2 Order of Accuracy and Basic Discretization

We have not yet mentioned how we intend to evaluate u at different Gaussian points. At the moment, we are only able to deﬁne a ﬁrst-order ﬁnite volume method based entirely on cell averages, which we call basic discretization. If vi denotes the cell average of our numerical solution (which still depends on time!), where we employ the discretization of the initial values as Z 1 vi (t = 0) = u0 (x, t) dσi , |σi | σi we can evaluate the numerical ﬂux function at exactly one Gaussian point – in the middle of ∂sik between the cell averages vi and vk . The parameter value of the middle point is ζν = 0, and the corresponding weight is given by ων = 2. Hence, the basic discretization is given by 2 ” d 1 XX “ sik g vi (t), vk (t); nik vi (t) = – . dt |σi | =1

(10.5)

k∈Ii

Note that this is exactly the same scheme given in (7.44) (rewritten for our boxes) if forward difference is applied for the time derivative. Concerning the spatial accuracy, we deﬁne order as follows. Our basic discretization (10.5) is said to be of order r if “ ” D E g vi , vk ; nik = f (v(xik (0), t)), nik + O(hr ) . Since we have only used one Gaussian point (nG = 1), the only possible values for r are 1 or 2. In order to decide on the accuracy question, we must examine the approximation properties of cell averages. From the deﬁnition of a barycenter, we know that Z 1 x dσi = ci (10.6) |σi | σi if ci are the coordinates of the barycenters of σi . Now let α = (α1 , α2 ) be a multiindex and consider the Taylor expansion of a smooth function u around the barycenter ci : u(x, t) =

r–1 X 1 X (x – ci )α ∂ α u|x=ci + O(|x – ci |r ) . μ! μ=0 |α|=μ

10.3 Higher-Order Finite Volume Schemes

Applying the cell average results in 1 |σi |

Z σi

u dσi =

Z r–1 X 1 X 1 (x – ci )α dσi ∂ α u|x=ci + O(|x – ci |r ) , μ! |σ | i σ i μ=0 |α|=μ

and choosing r = 2 yields ﬁ ﬂ Z Z 1 1 u dσi = u(ci , t) + (x – ci ) dσi , ∇x u(x, t)|x=ci + O(|x – ci |2 ) . |σi | σi |σi | σi R Due to (10.6), we ﬁnd that σi (x – ci ) dσi = 0, and we ﬁnally arrive at 1 |σi |

Z σi

u dσi = u(ci , t) + O(h2 ) .

Hence, the cell average operator represents the function u at the barycenter with second-order accuracy. However, from the computation above, it follows that, if we take another point xi ∈ σi which is not the barycenter, then Z 1 u dσi = u(xi , t) + O(h) |σi | σi holds. We can thus conclude that our basic discretization (10.5) is of ﬁrst-order spatial accuracy only.

10.3 Higher-Order Finite Volume Schemes

The analysis of accuracy given above can provide us with a hint about how to increase the spatial accuracy of ﬁnite volume methods. Suppose there were a smooth function (x, t) → pi (x, t) deﬁned for all x ∈ σi for which pi (x, t) – u(x, t) = O(hr ) for all x ∈ σi , as well as Z Z 1 1 pi dσi = u dσi . |σi | σi |σi | σi Then we could deﬁne a ﬁnite volume method as nG 2 “ “ ” “ ” ” 1 X X sik X d 1 ων g pi xik (ζν ), t , pk xik (ζν ), t ; nik vi = – . dt |σi | |σi | 2 ν=1 =1 k∈Ii

(10.7)

209

210

10 Finite Volume Models

We say that ﬁnite volume method (10.7) is of spatial order q if nG X

“ “ ” “ ” ” ων g pi xik (ζν ), t , pk xik (ζν ), t ; nik

ν=1

Z

1

= –1

D “ “ ”” E f u xik (ζ), t , nik dζ + O(hq ) .

It is immediately apparent that this deﬁnition generalizes that given for the basic discretization where we used nG = 1 and, correspondingly, ω1 = 2. It is then easy to show (see [20]) that a ﬁnite volume method (10.7) is of order q = min{r, 2nG }, where r is determined by “ “ ” “ ” ” D “ “ ”” E g pi xik (ζμ ), t , pk xik (ζμ ), t ; nik = f u xik (ζμ ), t , nik + O(hr ) . (10.8) We are now left with the question of how to derive functions pi on the boxes σi . This task is usually known as the recovery procedure. Before we look at recovery procedures, we will make a few remarks about timestepping schemes. We write any ﬁnite volume scheme in the abstract form d vi (t) = –Lh ({p(t)}) , dt where vi (t) is the cell average on box σi at time t, and p(t) represents the recovery polynomial to be computed. The operator Lh stands for the complete right hand side of a ﬁnite volume method. We will only discuss explicit methods in this context. However, it is worth noting that Meister has done groundbreaking work on implicit time-stepping schemes, which can be found in the literature. 157) The simplest explicit time-stepping method is the Euler forward difference: d vi (t + Δt) – vi (t) vi (t) = + O(Δt) . dt Δt This method is ﬁrst-order accurate in time and exhibits a small range of stability, so that in actual computations of ﬂuid ﬂow, the restriction on the time step Δt is often more restrictive than that predicted by the CFL condition. This said, it must also be admitted that the forward difference method is the ﬁrst choice when developing and testing a new method. Very popular in engineering circles is the classical four-step Runge–Kutta timestepping method, but it is not advisable to use it with the kinds of schemes we are after. This is because temporal errors from the Runge–Kutta scheme can turn a ﬁnite volume method with a spatial total variation diminishing operator Lh into a non-monotone scheme. Shu and Osher developed families of Runge–Kutta schemes which conform with the monotonicity of the spatial discretization. 158) We 157) See A. Meister: J. Comp. Phys. 140 (1998) 311–345 158) See C.-W. Shu, S. Osher: J. Comp. Phys. 77 (1988) 439–417, and (the second part of that work): J. Comp. Phys. 83 (1989) 32–78

10.4 Polynomial Recovery

can highly recommend the use of the second- or third-order variant, for which there is an economical form given by (0)

vi = vi (nΔt)

` ´ (1) (0) vi = vi – ΔtLh p(0) ` ´ 1 ` ´ 1 (2) (0) vi = vi – ΔtLh p(0) – ΔtLh p(1) 2 2 (2) vi ((n + 1)Δt) = vi

for the second-order variant and (0)

vi = vi (nΔt)

` ´ (1) (0) vi = vi – ΔtLh p(0) ` ´ 1 (2) (0) vi = vi – ΔtLh p(0) – 4 ` ´ 1 (3) (0) vi = vi – ΔtLh p(0) – 6 (3) vi ((n + 1)Δt) = vi

` ´ 1 ΔtLh p(1) 4 ` ´ 2 ` ´ 1 ΔtLh p(1) – ΔtLh p(2) 6 3 (k)

for the third-order scheme. Here we have used the notation vi to mark the stages of the time-stepping scheme. It is worth noting that, during the time stepping, one additional recovery process is required for the second-order scheme while two other recoveries must be performed in the third-order case. Since the recovery has to be performed for each box and is usually time-consuming, the second-order scheme may be the practical choice.

10.4 Polynomial Recovery

At the beginning of each new time step, we would like to construct on each box σi a polynomial pi of ﬁxed degree such that both 1 |σi |

Z σi

pi dσi =

1 |σi |

Z σi

u dσi ,

and pi (x, t) – u(x, t) = O(hr ) hold for each ﬁxed time t. Polynomials satifying these conditions are called recovery polynomials. They can always be written in the form pi (x, t) =

r–1 X 1 X (x – ci )α aα (t) , μ! μ=1

|α|=μ

(10.9)

211

212

10 Finite Volume Models

where ci = (c1,i , c2,i ) is again the barycenter of box σi ; i.e., a linear recovery polynomial (r = 2) would be pi (x, t) = a(0,0) + a(1,0) (x – c1,i ) + a(0,1) (y – c2,i ) and a quadratic recovery polynomial would read pi (x, t) = a(0,0) + a(1,0) (x – c1,i ) + a(0,1) (y – c2,i ) 1 1 + a(2,0) (x – c1,i )2 + a(1,1) (x – c1,i )(y – c2,i ) + a(0,2) (y – c2,i )2 . 2 2 It is now obvious that the coefﬁcients aα of a recovery polynomial (10.9) must satisfy the condition “ ” aα = ∂ α u|x=ci + O hr–|α| , since if u is smooth and can be expanded according to u(x, t) =

r–1 X 1 X (x – ci )α ∂ α u|x=ci + O(hr ) , μ! μ=1

|α|=μ

then pi (x, t) – u(x, t) =

r–1 X ` ´ 1 X (x – ci )α aα – ∂ α u|x=ci + O(hr ) . μ! μ=1 |α|=μ

The condition on the coefﬁcients then follows from |pi (x, t) – u(x, t)| u

r–1 X ˛ ˛ 1 X |x – ci ||α| ˛aα – ∂ α u|x=ci ˛ + O(hr ) μ! μ=1

|α|=μ

r–1 X ˛ 1 X |α| ˛˛ u h aα – ∂ α u|x=ci ˛ + O(hr ) . μ! μ=1 |α|=μ

It is clear that a recovery polynomial of degree 1 (r = 2) has two unknown coefﬁcients, while a polynomial of degree 2 (r = 3) has six unknown coefﬁcients. In general, in two spatial dimensions, the number of coefﬁcients to be computed for order r is given by the binomial coefﬁcient „ « r+1 . r–1 To compute the unknown coefﬁcients of a recovery “ ”polynomial (10.9) on box σi , one needs the cell average vi on σi , as well as r+1 r–1 – 1 further cell averages on boxes in the neighborhood of σi . A recovery algorithm must therefore search for neighboring boxes and then compute the polynomial.

10.4 Polynomial Recovery

Tin

i

σi

Fig. 10.4 Geometry for linear recovery.

We start with a simple but robust and accurate algorithm. Consider the box σi around node i, like the one in Fig. 10.4. Since the data is stored at the nodes of the triangulation, we can compute a linear polynomial on every triangle Tin sharing node i: πTin =

3 X

vm Φnm

m=1

where m m Φnm (x, t) = bm 0 (t) + b1 (t)x + b2 (t)y

is the Lagrangian interpolation polynomial deﬁned by Φnm (xl , t) = δli , where the xl denote the nodes of triangle Tin . Hence, we end up with as many linear polynomials πTin , n = 1, . . . , #T as there are triangles sharing node i (and which therefore constitute the box σi ). Note that these polynomials are interpolants of cell averages. A linear recovery polynomial should take the form pi (x, t) = a(0,0) + a(1,0) (x – c1,i ) + a(0,1) (y – c2,i ) , and since pi (ci , t) = vi (t), the value of a(0,0) is the cell average on σi . Hence, we can rewrite the polynomial as pi (x, t) = vi + ∇wi , x – ci ,

(10.10)

where we have written ∇wi = (a(1,0) , a(0,1) ) on σi . We can now use our linear interpolants on the triangles surrounding i and deﬁne Z ˛ 1 ∇x πTin ˛T n dσi . ∇wi := i |σi | σi

213

214

10 Finite Volume Models

Since the gradients ∇x πTin are constant on the triangles they belong to, we could also have written ∇wi =

1 X ∇x πTin |Tin ∩ σi | . |σi | n

This would indeed result in a linear recovery polynomial, but it would result in serious stability problems if applied to hyperbolic conservation laws. A cure is provided by direct slope limiting. The recovery polynomial (10.10) is augmented by a slope limiter Li : pi (x, t) = vi + Li ∇wi , x – ci . Several functions have been derived in the literature, and one that is widely used is that designed by Barth and Jespersen: 159) Li = min L∂σ ∂σik

ik

with

L∂σ

ik

j ﬀ 8 vik+ – vi > > min 1, ; > > pˆi – vi > > < j ﬀ v – – vi = min 1, ik ; > > > pˆi – vi > > > : 1 ;

pˆi – vi > 0 pˆi – vi < 0 pˆi – vi = 0

where vik+ = maxk∈Ii {vi , vk }, vik– = mink∈Ii {vi , vk }, and pˆi denotes the evaluation of the unlimited polynomial (10.10) at the Gaussian points on ∂σik . In practice, however, slight oscillations can still be seen in the computation of shock waves. The scheme can be made total variation diminishing if the resulting recovery polynomial on every box is also checked to make sure that it respects the monotonicity of the cell averages. An even simpler algorithm is given by the following recipe. As we did above, we compute the different Lagrange polynomials πTin and take their gradients. Then we choose ∇wi = min |∇x πTk | k

i

for the gradient of the recovery polynomial on box σi . We now want to describe an algorithm which is capable of recovering higherdegree polynomials. It was developed by Abgrall 160) and is exploited in [20]. As we 159) T.J. Barth, D.C. Jespersen: The design and application of upwind schemes on unstructured meshes (AIAA Paper 89-03660). Reston: AIAA 1989 160) R. Abgrall: J. Comp. Phys. 114 (1994) 45–58

10.4 Polynomial Recovery

Timin

Fig. 10.5 Three sets of triangles for quadratic recovery.

did above, we construct linear polynomials πTin on every triangle Tni sharing the node i via the ansatz X πTin = (x – ci )α anα |α|u1

and the coefﬁcients from Z 1 πT n dσl = vl , |σl | σl i where the index l runs over the three nodes of the triangle. Among all those triangles sharing node i, we denote the triangle for which 8 9 <X = X˛ ˛ n ˛aN ˛ = min |a | α α n : ; |α|=1

|α|=1

is valid as Ti min (TN i , say). If we wanted to work with a linear recovery polynomial, we could stop here and set pi = πTNi . Otherwise, if Ti min must be found, we consider three sets of three triangles in the vicinity of Ti min , as deﬁned in Fig. 10.5. For each of the three sets, we compute the quadratic polynomial X (x – ci )α aα |α|u2

R P from the condition σ1l σl |α|u2 (x – ci )α aα = vl , where l runs over the indices belonging to the nodes of the three triangles in the set under consideration. In each case, this gives ﬁve equations for the ﬁve unknown coefﬁcients. The sixth coefﬁcient, a(0,0) , is obviously the cell average vi . From the three quadratic polynomials, P we choose the one for which |α|=2 |aα | is the smallest as the recovery polynomial. We could now go on to enlarge the set of triangles, compute third-order polynomials, and so forth. However, the condition number for the linear systems gets worse very quickly. Finite-volume schemes with orders that are higher than three are therefore not found in the literature, except in the spatially one-dimensional case.

215

216

10 Finite Volume Models

10.5 Remarks Concerning Non-polynomial Recovery

The only reason for using polynomials in the recovery process is their simplicity. Therefore, other choices for recovery functions can be found in the literature. One starting point when choosing recovery functions is the theory of optimal recovery, as founded by Golomb and Weinberger 161) and exploited by Micchelli and Rivlin. 162) The theory applies in particular to reproducing kernel Hilbert spaces in which the point evaluation functional δx , deﬁned by means of δx u = u(x) , is continuous. It can, however, be applied to arbitrary Banach spaces in which the point evaluation functional is continuous. In ﬁnite-volume methods, we actually need to recover point values at the Gaussian points of the edges between neighboring boxes. It turns out that we can ﬁnd optimal functions in such spaces in the following sense. Consider the recovery problem on box σi , and assume that we use M–1 additional boxes σi1 , σi2 , . . . , σiM–1 in the vicinity of σi , so that the average data from M cells can be used. The given data consist of the cell averages for the M boxes. Let the space we want to explore be a Banach space V with a continuous point evaluation functional. We further assume the existence of a convex neighborhood U ⊂ V of 0 for which ∀u ∈ V∃c > 0 :

cu ∈ U

and u ∈ U ⇔ –u ∈ U holds. We shall come back to the meaning of this convex neighborhood later. The mapping ! Z Z Z 1 1 1 V u → Ii u = u dσi , u dσi2 , . . . , u dσiM–1 ∈ RM σ i σi σi1 σi2 σiM–1 σiM–1 is called an information operator. The linear functional V u → δx u = u(x) is called a property operator. Every (not necessarily linear) operator RM Ii u → Ri (x)Ii u ∈ R 161) M. Golomb, H.F. Weinberger: Optimal approximation and error bounds. In: R.E. Langer (ed.): On numerical approximation. Madison: Univ. Wisconsin Press 1959

162) C.A. Micchelli, T.J. Rivlin: Lectures on optimal recovery. In: P.R. Turner (ed.): Numerical analysis, Lancaster 1984 (Lecture Notes in Mathematics 1129:12–93). Berlin: Springer 1984

10.5 Remarks Concerning Non-polynomial Recovery

is a recovery operator. Since we want to recover the point values of a function u, we deﬁne the recovery error by ERi (x) (δx , Ii , U ) = sup |δx u – Ri (x)Ii u| . u∈U

The lower bound E(δx , Ii , U ) = inf ERi (x) (δx , Ii , U ) = inf sup |δx u – Ri (x)Ii u| Ri (x)

Ri (x) u∈U

is called the proper recovery error. A recovery operator R∗i (x) with the property ER∗i (x) (δx , Ii , U ) = E(δx , Ii , U ) is an optimal recovery operator. We have not used the convex neighborhood U around 0 ∈ V, but it does have an important meaning. The problem of recovery from cell averages is not solvable in the form in which we stated it. This can be seen as follows. Consider all of the linear functionals contained in Ii u to be linearly independent and linearly independent from the property operator. Under these assumptions, there is a function w ∈ V with δx w = 1 and ∀k ∈ {i, i1 , . . . , iM–1 } :

1 σk

Z σk

u dσk = 0 ,

otherwise the hyperplane deﬁned by δx w = 1 would be parallel to one of the hyperplanes deﬁned by the linear functionals in the information. The function u + ξw is an element of V for every u and ξ ∈ R, and it satisﬁes the equation Ii (u + ξw) = Ii u + ξIi w = Ii u . On the other hand, we have δx (u + ξw) = δx U + ξδx w = u(x) + ξ . Thus, arbitrary values of ξ result in arbitrary values for δx . In order to remedy this situation, the convex neighborhood U of 0 ∈ V comes into play. We need a linear restriction operator T , deﬁned by V u → T u ∈ V . Aided by this operator, we can deﬁne U = {u ∈ V | T uV u 1} , and it is then easy to show that U is the convex neighborhood sought. If we now ask for recovery in U instead of V, we do not encounter the problems described above

217

218

10 Finite Volume Models

any more, and we may be able to ﬁnd an optimal recovery function. We refer the reader to monograph [20], in which the theory of optimal recovery is fully developed for ﬁnite volume methods. The interesting case of reproducing kernel Hilbert spaces reveals that splines are the optimal recovery functions. A spline is deﬁned as a function that minimizes the norm (or semi-norm) in a Hilbert space. It is in this context that radial basis functions play an important role, like the thin-plate spline in the Beppo–Levi space Φ(x) =

M–1 X j=0

λj

1 σij

Z

y

σi j

|x – y|2(m–1) log(|x – y|) dσij + v(x) ,

where i0 = i and v is any function from the kernel of the semi-norm in the Beppo– Levi space. For details on the theory and implementation of radial basis functions, see [20].

10.6 Remarks Concerning Grid Generation

We cannot provide an overview of grid generation techniques for unstructured grids here; the reader should instead refer to the literature available on the subject. 163) In the case of triangulations in 2D, there are very useful adaptation algorithms for reﬁning or coarsening a given grid. We employ two different strategies to reﬁne a triangle, namely red reﬁnement and green reﬁnement, as shown in Fig. 10.6.

Fig. 10.6 Red (left) and green (right) reﬁnement. (Courtesy of Cambridge University Press) 164)

Red reﬁnements are preferable because they result in geometrically similar, smaller triangles, while green reﬁnement leads to small angles. Hence, green reﬁnement must only be used to ﬁt a region that was red-reﬁned to coarser triangulation in order to avoid hanging nodes. If certain patches of triangles around a node have been reﬁned they can also be recoarsened. Figure 10.7 shows a so-called resolvable patch which can be coarsened. Another resolvable patch is shown in Fig. 10.8. The complete algorithms, along with many examples of their use, are given in the literature. 165) 163) See, for example, J.F. Thompson, B.K. Soni, N.P. Weatherill (eds.): Handbook of grid generation. Boca Raton: CRC 1999

164) Figures 10.6–10.12 are taken from K.W. Morton, T. Sonar: Acta Num. 16 (2007) 155–238 165) See, for example, Chapter 3 in [15]

10.6 Remarks Concerning Grid Generation

Fig. 10.7 Resolvable patch (left) and recoarsened patch (right). (Courtesy of Cambridge University Press)

Fig. 10.8 Resolvable patch (left) and recoarsened patch (right). (Courtesy of Cambridge University Press)

Fig. 10.9 Grid and pressure distribution. (Courtesy of Cambridge University Press)

In Fig. 10.9, we show a grid in the vicinity of a NACA 0012 airfoil and the pressure distribution computed using a second-order ﬁnite-volume method. The angle of attack is α = 1.25◦ and the onﬂow Mach number Ma∞ = 0.8. Under these conditions, we expect a strong shock on the upper surface of the airfoil and a weak one on the lower side. Both shocks can be seen in the pressure distribution but they are smeared out. If an error indicator is used to mark the triangles to be reﬁned, we arrive at a reﬁned grid such as that shown in Fig. 10.10. The corresponding pressure distribution now shows nicely captured shocks. A very impressive case is shown in Fig. 10.11. Here, an inviscid ﬂow with Ma∞ = 3 arrives from the left and hits a forward-pointing step in a channel. The resulting ﬂow in Fig. 10.12 shows a nice pattern of shocks, a contact discontinuity and a rarefaction wave.

219

220

10 Finite Volume Models

Fig. 10.10 Grid and pressure distribution. (Courtesy of Cambridge University Press)

Fig. 10.11 Adapted grid for channel ﬂow. (Courtesy of Cambridge University Press)

Fig. 10.12 Density distribution of channel ﬂow. (Courtesy of Cambridge University Press)

One question that we are yet to address is the selection of an error indicator. In order to get true control over the error, we need reliable information about the function space of solutions of the Euler or Navier–Stokes equations. This is presently not the case. However, a lot of work has recently been done on hyperbolic systems, and this can be found in the literature. 166)

166) See, e.g., P. Houston, J. Mackenzie, E. Süli, G. Warnecke: Num. Math. 82 (1999) 433–470; T. Sonar, E. Süli: Num. Math. 78 (1998) 619–658

221

Index a

c

adiabatic exponent 6 advection equation 13 ampliﬁcation factor 142 ampliﬁcation matrix 143 angle of attack 18 angular velocity vector 15 antidifference operator 134 approximate solutions 116 Archimedes 30 Archimedes’ principle 30 artiﬁcial time 90 asymptotically closed 117 asymtotically regular 117 Aw, A. 77 Aw–Rascle model 77

carburetor 16 Cauchy–Riemann equations 20 cell average 130 centered waves 63 CFL condition 114, 122, 136, 140, 159 – stronger 161 Chapligin solutions 28 Chapligin, C.A. 28 characteristic cone 76 characteristic ﬁeld 170 – genuinely nonlinear 170 – linearly degenerated 170 characteristics 10–12, 41 – numerical 135 circulation 15 Clausius, R.J.E. 49 closing of disturbed laminar ﬂow 102 congestion 38 congestion velocity 47 conservation laws 7 conservation of – energy 1, 5 – mass 1, 3 – momentum 1, 3 conservation principles xv, 1 conservative variables 5 consistency condition 138 consistency of a numerical method 122 consistency of numerical methods 133 contact discontinuity 45, 171 continuity equation 3 continuous convergence 119 – of relations 151 contravariant components of a vector 190 contravariant metric tensor 192

b back side of a shock 45 basic discretization 208 Bernoulli equation 16 – generalized 17 Bernoulli, Daniel 16 Bois-Reymond, P. du 119 Boltzmann constant 106 Boltzmann, L. 49 boundary layer 95 boundary layer equations 99 boundary layer theory 89 box 206 Brandt, G.B. 106 Breuss, M. 71 Briggs, H. 131 Buchanan, J.E. 144 Buckley–Leverett equation 71 buoyancy – dynamic 14 – static 26, 30 Burgers’ equation 10 Burgers, J. 10

222

Index control volume 206 convection term 5 convergence of numerical methods 74, 117 Courant, R. 114, 119, 135 covariant basis vectors 190 covariant components of a vector 191 covariant metric tensor 192 critical Reynolds number 89, 104 curve of neutral stability 104

d density 1, 7 density of momentum 3 difference equation 128 difference operator 127, 138 – central 132 – forward 131 difference stencil 135 diffusion term 14 discontinuity – admissible 170 discontinuous Galerkin method 129 discrete entropy conditions 119 discrete Fourier transform 141 discretely compact 117 divergence theorem 4 drag force 85

e e9 -rule 104 elements of FEM 128 elliptic grid generation 202 energy – interior 5 – kinetic 5 – total 5 Engquist, B. 124 Engquist–Osher scheme 124 ENO schemes 148 entropy 7, 49 entropy condition 50, 52 entropy ﬂux 53, 121 – numerical 123 entropy functional 53 entropy functions 121 entropy inequalities 68 entropy inequality 163 entropy solution 50, 53, 57, 62, 67, 68, 70, 118, 121, 163 – discrete 119, 123 equation of state 2, 7

equivalence theorem 139 Euler equations 1, 5 – entropy formulation 169 Euler’s law of vanishing momentum 80 Euler, L. 5, 80, 131 evolution equation 137 evolution equation for cell averages 130 evolution operator 137 – discrete 138 exact difference formula 132 existence proof for entropy solutions 113 explicit difference equation 128

f Fellehner, St. 91 ﬁnite difference grid 128 ﬁnite difference method (FDM) 73, 127 ﬁnite element method (FEM) 127 ﬁnite volume method (FVM) 127 First law of thermodynamics 5 ﬂow – barotropic 16 – incompressible 7 – irrotational 16 – isentropic 52 – laminar 85 – nonstationary 8 – plane 18 – potential 14, 15 – similar 85 – stationary 8 – steady-state 8 – supersonic 18 – through porous media 71 – turbulent 85 – subsonic 18 ﬂow separation 95 ﬂuid – Newtonian 81 – ideal 4 – inviscid 4 – real 4 – viscous 4 ﬂux 7 – numerical 122 ﬂux function – numerical 130 ﬂux limiter method 147 ﬂux splitting 123

Index ﬂux splitting scheme 124 force – conservative 16 Friedrichs, K.-O. 114, 135 front side of a shock 45 functional analysis of discretization algorithms 115 fundamental diagram 38 FVM on unstructured grids 148

inﬁnitesimal generator 137 iterative procedure 91

j Jacobian matrix 7 Jagger, S.F. 106 jam concentration 38 jet stream 16 jump condition 44

g

k

gas

Kantorowitch, L.W. 114 Kelvin’s theorem 17 Kelvin, Lord of Largs 17 Keyﬁtz, B. 125 Kreiss matrix theorem 144 Kreiss, H.-O. 144 Kruzkov solution 70 Kruzkov, S.N. 69, 121 Kutta, M.W. 26 Kutta–Zhukovsky buoyancy formulas 26 Kuznetsov, N.N. 59

– γ- 7 – ideal 4 Gasser, I. 106 genuine solution 137 genuinely nonlinear 63 Glimm, J. 114 global discretization error 140 Godunov method – for Euler equation 185 Godunov scheme 160 Godunov, S.K. 74 Greenshields’ model 39 grid 73 grid generation 149, 196 grid generation algebraic 198 grid generation analytic 198 grid generation cycle 202 grid of C-type 196 grid of H-type 197 grid of O-type 197 grid quality 197 grid topology 196 grid transformation 188

h Hagen, G.H.L. 89 Hagen-Poiseuille law 88 Harriot, T. 131 Harten, A. 121, 148 heated gas ﬂow 104 Hilgenstock 201 hodograph equation 28 hodograph plane 27 hydrodynamic paradoxon 16 hydrostatic equation 29 hydrostatic paradox 29 Hyman, J.M. 121

l Lahmann, J.-R. 104 Laurent, P.A. 23 Lax entropy solution 55 Lax shock conditions 68 Lax, P. 53, 121 Lax–Friedrichs scheme 155 Lax–Richtmyer theory 136 Lax–Wendroff scheme 147 Lax-Friedrichs method 164 Lax-Friedrichs scheme 146 Lax-Richtmyer theory 139 Lax-Wendroff theorem 125 Lea, C.J. 106 Legendre transformation 27 Lei, J. 126 Leibniz, G.W. 131 LeVeque, R.J. 125 Lewy, H. 114, 135 lift 26 Lighthill, M.J. 39 Lighthill–Whitham model 39, 77 limiter 148 local speed of sound 32

m i implicit difference equation 128 incremental form 157

Mach number 18, 176 Mach’s angle 19 Mach’s cone 19

223

224

Index Mach’s net 19 Mach, E. 18 main theorem of thermodynamics – ﬁrst 5, 49 – second 49 material time derivative 5 mathematical model xiii maximum principle 158 method of – characteristics 19 – hodographs 27 method of artiﬁcial compressibility 90 method of conservation form 122, 146 modiﬁed equation 144 monotone schemes 124 monotonicity preserving 156 Morton, K.W. 144 multistep method 74 Murman–Courant–Isaacson–Rees method 160

n Napier, J. 131 Navier, C.L.M.H. 83 Navier–Stokes equations 83 Neumann, J. von 141 Newton’s second law 3 Nickel, K. 95 no-slip condition 26, 106 non-standard conservation law 71 numerical domain of dependence 135 numerical ﬂux function 155 numerical viscosity coefﬁcient 156

o Oleinik solution 58 Oleinik, O. 58 one-step method 74 optimal recovery 216 order of a method 138 order of a numerical procedure 133 Orr–Sommerfeld equation 103 Osher, St. 124

p Panov, E.Y. 70 parameter space 187 partial slip 26 Petrov–Galerkin schemes 129 phase speed of perturbations 104

physical space 187 Plum, M. 104 Poise 83 Poiseuille, J.L.M. 89 polar coordinates 187 polymer 72 Prandtl number 85 Prandtl, L. 89, 95 pressure 6 – dynamic 16 – static 16 – total 24 primary grid 206 primitive variables 5 pseudospectral method 131

q quasilinear systems of partial differential equations 7

r Rankine–Hugoniot condition 42, 44, 170 Rankine-Hugoniot condition 76 rarefaction wave 41, 68, 172 Rascle, M. 77 reconstruction of functions 74 recovery polynomial 211 Reynolds averaging 102 Reynolds number 84 – critical 89, 104 Reynolds’ transport theorem 2, 104 Reynolds, O. 2 Richtmyer, R.D. 114, 144 ride impulse 41 Riemann invariant 68 Riemann invariants 170 Riemann problem 39, 73 – complete solution 178 – local 160 Riemann solvers 74 Riemann, B. 39 Rinow, W. 120 round-off errors 140

s scaled quantities 107 scheme – (2k+1)-point 122, 146 – 3-point 135, 146 – upwind 136 Schlichting, H. 97 second law of thermodynamics 49

Index secondary grid 206 semigroup property 137 shallow water equations 35 shift operator 131 shift operator – backward 141 – forward 142 shock 42, 45, 68, 171 shock conditions 68 shock curve 45 shock Mach number 176 similarity solution 171 slope limiting 214 smooth connection of states 66 solution operator 137 – generalized 137 Sommerfeld, A. 103 sound propagation 29 sound velocity, local 9, 32 source terms 200 spatial order of a FVM 210 speciﬁc heats 7 speciﬁc thermal conductivity 85 spectral collocation method 131 spectral discretizations 131 spectral element method 131 spectral Galerkin method 131 stability – numerical 114, 139 – of laminar ﬂows 102 – spatial 103 – temporal 103 stability criterions 143 step size 74 Stokes approximation 89 Stokes, G.G. 83 stream function 20 streamline 9 streamline diffusion methods 129 stress tensor 80 strictly hyperbolic 9 Struckmeier, J. 106 Stummel, F. 114 sucking off of the boundary layer 99 summation by parts 134 symbol of a shift operator 141

t Tadmor’s lemma 165 temperature 7 test elements 70, 116 test function 35

test functions 128 thermal conductivity 105 time step size 73, 122 Tollmien, W. 95 Tollmien-Schlichting waves 103 trafﬁc ﬂow 37 trafﬁc jam 46 transformed divergence term 195 transformed Laplace equation 195 transport theorem 2 tribology 89 truncation error 74, 133, 138 tunnel ﬁres 106 TVB schemes 124 TVD property 156 TVD schemes 124 Tveito, A. 71

u unscaled quantities 107 unstructured grids 130, 148, 218

v Vainikko, G. 115 variables – conservative 5 – primitive 5 velocity potential 15 – complex 20 viscosity 4, 72 – kinematic 83 viscosity coefﬁcient 155 – ﬁrst 83 – second 83 viscosity form 156 von Neumann stability criterion 144 vortex line 15 vorticity vector 15

w Wang, C.Y. 89 weak formulation of a numerical method 116 weak solution 13, 35, 67, 116 Wendroff, B. 125, 147 Whitham, C.B. 39 wing theory 89 Winther, R. 71

z Zhukovsky, N.J. 26

225

227

Suggested Reading The aim of this book is to provide an introduction to all of the aspects of mathematical ﬂuid dynamics (MFD); it would be impossible to cover all of these aspects in depth in this one book. However, many excellent, modern and extensive monographs on the subject are available. Most of them describe particular areas of MFD (like numerical procedures), they treat the basic equations theoretically, or they justify the mathematical treatment from a physical or engineering point of view. Some modern books that are useful for more detailed studies are listed here: 1 Brower W.B., Dynamics of ﬂows in one space dimension, Boca Raton: CRC, 1999 2 Chorin A.J., Marsden J.E., A mathematical introduction to ﬂuid mechanics, 2nd ed, New York: Springer, 1984 3 Dubois T., Jauberteau F., Temam R., Dynamic multilevel methods and the numerical simulation of turbulence, Cambridge: Cambridge Univ., Press, 1999 4 Emanuel G., Analytic ﬂuid dynamics, Boca Raton: CRC, 2001 5 Feistauer M., Mathematical methods in ﬂuid dynamics, Boca Raton: CRC, 1993 6 Ferziger J.H., Peric M., Computational methods for ﬂuid dynamics, Berlin: Springer, 1999 7 Freistühler H. (ed), Analysis of systems of conservation laws, Boca Raton: Chapman&Hall/CRC, 1998 8 Frisch U., Turbulence: the legacy of A.N. Kolmogoroff, Cambridge: Cambridge Univ. Press, 1995 9 Godlewski E., Raviart P.-A., Numerical approximation of hyperbolic systems of conservation laws, Berlin: Springer, 1996 10 Holmes P., Lumley J.L., Berkooz G., Turbulence, coherent structures, dynamical systems and symmetry, Cambridge: Cambridge Univ. Press, 1996

11 Kröner D., Numerical schemes for conservation laws, Chichester: Wiley Teubner, 1997 12 LeVeque R.J., Numerical methods for conservation laws, Basel: Birkhäuser, 1990 13 Majda A., Compressible ﬂuid ﬂow and systems of conservation laws in several space variables, Berlin: Springer, 1984 14 Malek J., Necas J., Rokyta M., Advanced topics in theoretical ﬂuid mechanics, Boca Raton: CRC, 1999 15 Meister A., Struckmeier J., Hyperbolic partial differential equations, Braunschweig: Friedr. Vieweg, 2002 16 Richtmyer R.D., Morton K.W., Difference methods for initial value problems, 2nd ed, New York: Wiley, 1967 17 Schlichting H., Gersten K., Boundarylayer theory, 8th ed, Berlin: Springer, 2000 18 Serre D., Systems of conservation laws 1, Cambridge: Cambridge Univ. Press, 1999 19 Smoller J., Shock waves and reactiondiffusion equations, Berlin: Springer, 1983 20 Sonar T., Mehrdimensionale ENOVerfahren, Stuttgart: B.G. Teubner, 1997 21 Spurk, J.H., Fluid mechanics, Berlin: Springer, 1997 22 Warsi, Z.U.A., Fluid dynamics, theoretical and computational approaches, Boca Raton: CRC, 1998